Local PI Theory of Jordan Systems II

Fernando Montaner^{∗}

Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain

Abstract: We pursue the study initiated in a previous paper of Jordan systems having nonzero local algebras that satify a polynomial identity. We define the extended centroid of a nondegenerate Jordan system, the corresponding central extension, which we call the extended central closure, and prove a Jordan analogue of Martindale’s Theorem on prime algebras having a generalized identity: If J is anondegenerate Jordan system with nonzero PI-elements, then the extended central closure of J has nonzero socle, equal to its PI-ideal.

Introduction

This paper is a continuation of [24], where we started the study of Jordan systems having nonzero local PI-algebras. This condition has repeteadly arised in several papers on Jordan theory in a more or less explicit form, beginning with Zelmanov’s work on triple systems and pairs, and on Goldie’s theorems for Jordan algebras. That condition also appears in Anquela, Cortés and Montaner’s work on primitive Jordan algebras. More recently, D’amour and McCrimmon’s extension [2] to quadratic systems of Zelmanov’s results has revealed the central role of local algebras and, in particular, local PI-algebras, in Jordan theory. This has been exploited by Anquela and Cortés [3] in their study of primitive Jordan systems, which based on the local characterization of primitivity.

As commented in our previous paper [24], the present work was motivated by the on-going project of extending to quadratic Jordan algebras Zelmanov’s Goldie’s theory for linear Jordan algebras. This requires the comparison of inner ideals of special Jordan algebras and one-sided ideals of their associative envelopes (to deal with notions like uniform dimension or singularity), and here the non-existence of PIelements is essential. Thus, a structural result was needed to deal with the presence of nonzero PI-elements.

A different motivation cames from the classification of strongly prime Jordan systems. There, as in Jordan algebras, the presence of some particular classes of identities is of fundamental structural importance [2]. So that situation suggest the interest of the study of general PI-Jordan systems. There is, however, an important feature in the kind of identities that are significant in that classification, they are homotope polynomial identities, i.e. identities (of algebras) which hold in every homotope (hence in every local algebra) of the system. This, of course, leads again to the study of local PI algebras.

∗ Partially supported by the DGICYT, PB94-1311-C03

This twofold motivation suggests two sources of inspiration, associative PI-theory for the latter, and associative GPI-theory for the first. The Relation with this second theory becomes apparent if one realizes that the presence of a GPI in a semiprime associative algebra is equivalent to the existence of a PI one-sided ideal, and this is equivalent to the existence of a nonzero local PI-algebra (see [25]). Therefore, from a srtuctural viewpoit, the results that should be extended to Jordan systems are Kaplansky’s and Amitsur’s theorems on primitive PI and GPI algebras, respectively, and Posner’s and Martindale’s theorems on prime PI and GPI algebras respectively.

In our previous paper [24] we defined the set PI(J) of PI-elements of J (those which produce a PI local algebra), proved that it is an ideal if J is nondegenerate, and studied primitive algebras with PI(J) ≠ 0. The main result about these systems was an analogue of Amitsur’s theorem on GPIs, asserting that they have nonzero socle, equal to the set of PI-elements. From this one gets an analogue of Kaplansky’s theorem saying that homotope-PI primitive Jordan systems are simple with capacity.

In the present paper we address the corresponding theorems in the prime case. Our main result is theorem 5.1, which asserts that a strongly prime Jordan system with nonzero PI-elements has a primitive extended central closure.

The paper is organized as follows. After a section devoted to preliminaries, we define in section 1 the extended centroid of a nondegenerate Jordan system. The construction is based on the non-associative extension of the extended centroid given in [12, 9], although we have to surmount the difficulties inherent to our quadrating setting. In section 2 we define the scalar extension naturally attached to the extended centroid, which we call the extended central closure (to distinguish it from the – ordinary– central closure), and, in section 3 we prove some of its properties. Here, the results are inspired by the idea of relating “central quotients” to localizations in certain categories, as expressed in [28]. We point out, however, that the nonlinearity of Jordan systems only allows a rather rudimentary version of those ideas. We point out, also, that there are several precedents [10, 11] of the use of extended centroids in (linear) Jordan theory.

Section 4 is devoted to the proof of our main theorem, which we base on an ultrafilter argument inspired by the one originally used by Martindale [19]. Finally, in section 5 we specialize our result to homotope-PI algebras, to obtain an analogue of Posner-Rowen’s theorem for Jordan systems.

0. Preliminaries

Throughout Φ will be a fixed unital commutative ring.

0.1 We will work with Jordan pairs, triple systems, and algebras over Φ. Our main sources of notation, termonology, and basic results are [15,16,23]. We record in this section some of those notations and results.

A Jordan algebra J has products x^{2} and U_{x}y, quadratic in x and linear in y, whose linearizations are

x ∘ y = V_{x}y = (x + y)^{2} − x^{2} − y^{2},
U_{x,y}z = V_{x,z}y = {x, z, y} = U_{x+y}z − U_{x}z − U_{y}z.

A Jordan Pair V = (V^{+}, V^{−}) has products Q_{x}y for x ∈ V^{σ} and y ∈ V^{−σ}, σ = ±, with linearizations Q_{x,z}y = D_{x,y}z = {x, y.z} = Q_{x+z}y − Q_{x}y − Q_{z}y. We also mention the important Bergmann operators given by B_{x,y}z = z − D_{x,y}z + Q_{x}Q_{y}z for x, z ∈ V^{σ}, y ∈ V^{−σ}, which satisfies Q_{B_{x,y}z} = B_{x,y}Q_{z}B_{y,x}.

A Jordan triple system T has product P_{x}y whose linearizations are P_{x,y}z = L_{x,z}y = {x, z, y} = P_{x+y}z − P_{x}z − P_{y}z. The Bergmann operator in triple systems is given by B_{x,y}z = z − L_{x,y}z + P_{x}P_{y}z.

A Jordan algebra gives rise to a Jordan triple systems with P = U. If a Jordan triple systems has an element 1 with P_{1} = Id, the identity, then it is a unital Jordan algebra with square x^{2} = P_{x}1.

We denote by Γ(J) the centroid of a Jordan system J. This is a reduced commutative ring if J is nondegenerate, a domain actig faithfully on J if J is strongly prime, and a field if J is simple.

0.2 Doubling a Jordan triple system T produces a Jordan pair V (T) = (T, T) with Q_{x}y = P_{x}y. Reciprocally, each Jordan pair V = (V^{+}, V^{−}) gives rise to a polarized triple system T (V) = V^{+} ⊕ V^{−} with product P_{x^{+}⊕x^{−}}y^{+} ⊕ y^{−} = Q_{x^{+}}y^{−} ⊕ Q_{x^{−}}y^{+}. Niceness conditions such as nondegeneracy, primeness, strong primeness and others are inherited by the polarized triple system of a Jordan pair. However this does no longer hold in the reverse direction, from Jordan triple systems to their double Jordan pairs. To remedy that situation, D’Amour and McCrimmon [1, p. 229], and Anquela and Cortés [3, p. 667] defined tight doubles:

Given a Jordan triple system T, a tight double of T is a quotien pair V (T)/I = (T/I^{+}, T/I^{−}) where I is an ideal (I^{+}, I^{−}) of V (T) which is maximal with respect to I^{+} ∩ I^{−} = 0 (so that the I^{σ} are semi-ideals of T, but the may not be ideals). These always exist and share niceness properties with T (see 5.2 and 5.3 of [3]).

0.3 Since at some point we will make use of an utltraproduct argument involving systems with nonzero socle, it will be important for us to be able to express strong primeness by means of an elementary sentence, so that we can pass it from the factor to the ultraproduct. Such a characterization was found in [5]:

– A Jordan pair V = (V^{+}, V^{−}) is elementary prime if for any pair of elements x, y ∈ V^{σ}, σ = ±, Q_{x}Q_{V^{−σ}}Q_{y} = 0 implies x = 0 or y = 0.

– A Jordan algebra J is elementary prime if for any x, y ∈ J, U_{x}U_{J}U_{y} = 0 implies x = 0 or y = 0.

– A Jordan triple system T is strongly prime if for any x, y ∈ T, P_{x}P_{T} P_{y} = 0 and P_{x}P_{T} P_{T} P_{y} = 0 imply x = 0 or y = 0.

By the results of [5], strong primeness is equivalent to elementary primeness. Although in its general form these results make use of the classification theorems of [2], as we have mentioned, we will only need these facts for nondegenerate systems with nonzero socle. Since the results of [2] are only needed in [5] to reduce the problem to a separate study of hermitian systems and forms of systems with nonzero socle, we can make use of the results concernig the latter, which are independent of [1], to state:

0.4 Theorem. Le J be a nondegenerate Jordan system with nonzero socle. Then J is strongly prime if and only if it is elementary prime.

Proof: That any elementary prime prime Jordan system is strongly prime is straightforward (1.3 of [5]).

For the reciprocal, if our system J is a Jordan pair, then Soc(J_{λ}) is strongly prime, hence simple [17]. Then, by 1.9 of [5], Soc(J) is elementary prime, and J itself is elementary prime by 1.6 of [5], and none of these results depends on [2].

Now, let J be a Jordan triple system or an algebra, and W be a tight double. Then W is strongly prime (by 5.2 of [3]), hence elementary prime. If W = (J, J) is already tight, then J is elementary prime.

If (J, J) is not tight, Soc(J) = Soc(T (W)) through the natural embedding J ⊆ T (W) (see 4.2(d) of [24]). Thus Soc(J) is elementary prime by 1.9 and 2.6 of [5], and J is elementary prime by 2.8 of [5]. Notice that this is again independent of the results of [2].

0.5 Let (V^{+}, V^{−}) be a Jordan pair and a ∈ V^{σ}, where σ = ±. The a-homotope of V, denoted by (V^{−σ})^{(a)}, is the Jordan algebra over the Φ-module V^{−σ} with operations U_{x^{−σ}}^{(a)}y^{−σ} = Q_{x^{−σ}}Q_{a}y^{−σ} and (x^{−σ})^{2} = Q_{x^{−σ}}a.

The set Ker a of all x^{−σ} ∈ V^{−σ} such that Q_{a}x^{−σ} = Q_{a}Q_{x^{−σ}}a = 0 is an ideal of a, so that the quotient V_{a}^{−σ} = (V^{−σ})^{(a)}/Ker a is again a Jordan algebra. This is called the local algebra of V at a.

For triple systems and Jordan algebras, homotopes and local algebras are defined in the same way, just delete the superscripts σ from the previous definitions. We refer to [1] for a thorough study of local algebras.

Local algebras can be considered as particular cases of subquotients [24, 0.4] as defined by Loos and Neher [18], so the general results of that theory apply to them. In particular (see [18, 2.7(b) and (c)]),

0.6 Lemma. Let J be a nondegenerate Jordan system, a an element from J, and p : V → J_{a} the canonical map. Then Soc(J_{a}) = p(Soc(J)), and if Soc(J) is simple, so is Soc(J_{a}).

As a consequence, Soc(J_{a}) ≠ 0 for a Jordan system J, implies Soc(J) ≠ 0 (see [1]).

0.7 We refer to [14, 30, 1] for the basic notions on primitivity of Jordan systems. Recall that primitive Jordan systems are the building blocks of a structure theory based on the Jacobson radical (Theorem 5.4 of [14], Lemma 6 of [30] and Theorem 8 of [31]):

0.8 Theorem. For a Jordan system J, J/Soc(J) is a subdirect product of primitive Jordan systems.

Unlike for Jordan algebras, primitivity of Jordan triple systems and pairs is defined at a particular element (although it has been proved in [4] that primitivity can be moved to any other element). It is then natural to study primitivity of Jordan systems through their local algebras (see [1, 3, 4]). To do that, the key fact is that primitivity flows from pairs to their local algebras and back [3]. We will only make use of one direction (since the other relies on the structure theory [30, 2]), which we record in the following global-to-local inheritance theorem (Lemma 9 of [30] and Theorem 6.1 of [1]):

0.9 Theorem. If a Jordan pair V is primitive at b ∈ V^{σ}, then the local algebra V_{b}^{−σ} is primitive.

For algebras, the situation is neater since primitivity does not happen at a particular element (and since we do make use of the structure theory for algebras). In particular we have (Theorem 4.1 of [8]):

0.10 Theorem. Let J be a Jordan algebra, and o ≠ a ∈ J. If J is primitive, the J_{a} is primitive.

As mentioned before, our primitive systems will mainly be strongly prime with nonzero socle. For these on has (4.3 of [24]):

0.11 Proposition. Let J be a strongly prime Jordan system. If J has nonzero socle, then

(i) J is primitive (at each element of the socle if J is a pair or a triple system),

(ii) If J is an algebra or pair, then each local algebra is primitive with nonzero socle,

(iii) If J is a triple system, then each local algebra is either primitive with nonzero socle, or a subdirect sum of two primitive algebras with nonzero socle.

0.12 We finally mention some facts from Jordan PI-theory. Recall that a polynomial f(x_{1}, …, x_{n}) ∈ FJ[X], the free Jordan algebra on the set X, is called essential if its image in the free special Jordan algebra SFJ[X] under the natural homomorphism has a monic leading term. A PI-Jordan algebra is a Jordan algebra which satisfies some essential f(x_{1}, …, x_{n}). There are Jordan analogues of Kaplansky’s Theorem and Posner’s Theorem (see [7]):

0.13 Theorem. Let J be a strongly prime Jordan algebra. If J is PI, then the central closure Γ^{−1}J is simple with capacity.

0.14 The operant notion of Jordan PI triple system or pair, is that of homotope-PI triple system or pair. We will use the notations of [1] and [3]. In particular, if f(x_{1}, …, x_{n}) is a polynomial in the free Jordan algebra FJ[X] on a countable set of generators X, and z is an element of the free Jordan triple system FT[X], the polynomial

f(z; x_{1}, …, x_{n}) = f^{(z)}(x_{1}, …, x_{n})

is the image of f under the homomorphism FJ[X] → FT[X]^{(z)} which is the identity on X.

A Jordan triple system T satisfies a homotope polynomial identity (homotope-PI, for short) if there is a polynomial f(x_{1}, …, x_{n}) in FJ[X] whose image in the free special Jordan algebra SJ[X] has a monic term of highest degree and such that the polynomial f(y; x_{1}, …, x_{n}) with y ∈ X different from the x_{i}, vanishes under all substitutions of elements y, x_{i} ∈ T. This definition extends to Jordan pairs V by considering their associated triple system T (V).

0.15 The fact that a Jordan system J satisfies a homotope-PI means that all homotopes, hence all local algebras, satisfy a given identity. We are interested in a weaker assertion, the existence of some a ∈ J for wich the local algebra J_{a} is PI. We call such an element a PI-element, and write PI(J) for the set of PI-elements of J (PI(V) = (PI(V^{+}), PI(V^{−})) if J = V = (V^{+}, V^{−}) is a Jordan pair). Thus, the fact that J has a nonzero PI-element can be abreviated to PI(J) ≠ 0. Since the present work continues [24], we recall here the main results of that paper.

0.16 Theorem. Let J be a nondegenerate Jordan system. Then PI(J) is an ideal of J.

We say that a Jordan system J is rationally primitive if it is primitive and has a nonzero PI-element. This is the Jordan analogue of strongly primitive associative algebras. Rational primitivity is characterized in the following analogue of Amitsur’s Theorem on generalizad identities

0.17 Theorem. Let J be a Jordan system. The following are equivalent

(a) J is rationally primitive,

(b) J is strongly prime and Soc(T) = PI(T) ≠ 0

(c) J is strongly prime and the local algebra at some nonzero element is a simple unital PI-algebra.

As a consequence one has an analogue for Jordan systems of Kaplansky’s theorem, with homotope polynomial identities on Jordan systems playing the role of polynomial identities on algebras.

0.18 Theorem. Let J be a primitive Jordan pair or triple system.

(i) If the local algebra at each element of J is PI, then J is simple equal to its socle.

(ii) If J satisfies a homotope-PI, then J is simple with capacity.

1. The extended centroid

In this section we define the extended centroid of a nondegenerate Jordan system. The construction is patterned after the corresponding construction in associative theory and its non-associative generalizations [12, 9]. The main diference is that the scalarity conditions in Jordan systems are not linear (since the operations are not linear), so in that respect, the situation is much the same as in the definition of the centroid of a Jordan system (cf. [16]).

1.1 Let T be a Jordan triple system and I an ideal of T. A linear mapping f : I → T will be called a T -homomorphism if for all y ∈ I, x, z ∈ T it satisfies:

i) f(P_{x}y) = P_{x}f(y),

ii) f(P_{I}T) ⊆ I, and f^{2}(P_{y}x) = P_{f(y)}x,

iii) f({y, x, z}) = {f(y), x, z}.

The set of T -homomorphisms with domain I will be denoted Hom_{T} (I, T).

For a Jordan pair V = (V^{+}, V^{−}) and an ideal I = (I^{+}, I^{−}), we define a V - homomorphism as a pair f = (f^{+}, f^{−}) of linear mappings f^{σ} : I^{σ} → V^{σ}, σ = ±, satisfying, for all y^{σ} ∈ I^{σ}, x^{σ}, z^{σ} ∈ V^{σ},

i’) f^{σ}(Q_{x^{σ}}y^{−σ}) = Q_{x^{σ}}f^{−σ}(y^{−σ}),

ii’) f^{σ}(Q_{I^{σ}}V^{−σ}) ⊆ I^{σ}, and f^{σ}(Q_{y^{σ}}x^{−σ}) = Q_{f^{σ}(y^{σ})}x^{−σ}),

iii’) f^{σ}({y^{σ}, z^{−σ}, x^{σ}}) = {f^{σ}(y^{σ}), z^{−σ}, x^{σ}}.

We denote by Hom_{V} (I, V) the set of V -homomorphisms from I into V.

For a Jordan algebra J, and an ideal I, an algebra J-homomorphism, is a Jhomomorphism for the underlying triple system which, in addition to i, ii and iii, satisfies

iv) f(y^{2}) ∈ I, and f^{2}(y^{2}) = f(y)^{2},

v) f(x ∘ y) = x ∘ f(y).

That is, f is a Ĵ-homomorphism for the underlying triple system of a unital hull Ĵ of J. Again we denote the set of J-homomorphisms defined on I by Hom_{J}(I, J).

Following [12] and [9], a pair (f, I), where f ∈ Hom_{J}(I, J) for a Jordan system (pair, triple or algebra) will be called a permissable map if the ideal I is essential. We will be interested in nondegenerate Jordan systems, and for those, an ideal I is essential if and only if it has zero annihilator.

1.2 We will define the extended centroid of a nondegenerate Jordan system as a set of equivalence classes of permissable maps modulo a suitable equvalence relation. For linear algebras [12, 9] this set has a ring structure coming from the addition of homomorphisms and from the composition of restrictions of homomorphisms. Defining the corresponding operations in our case requires some preparatory work. In fact we do not even know yet if the sets of homomorphisms introduced above are submodules of the usual Hom_{Φ}(I, J), and, moreover, J-homomorphisms do not always restrict to J-homomorphisms on smaller ideals (cf. below).

We will deal with these problems for Jordan triple systems, and later on we will reduce the pair and algebra cases to that one.

For permissable maps (f, I) and (g, L) of the Jordan triple system T, define a relation “∼” by:

(f, I) ∼ (g, L) if there is an essential ideal K of T, contained in I ∩ L, such that f(x) = g(x) for all x ∈ K.

It is easy to see that this is an equivalence relation. The quotient set C(T) will be called the extended centroid of T. We will write [f, I] for the equivalence class of the permissable map (f, I).

1.3 Lemma. Let T be a nondegenerate Jordan system, I an ideal of T and f, g ∈ Hom_{T} (I, T). Then, for all x ∈ I and y ∈ T, fgP_{x}y = gfP_{x}y.

Proof: Notice first that g(P_{x}y) and f(P_{x}y) belong to I, hence fgP_{x}y and gfP_{x}y make sense. We will show that [f, g]P_{x}y = fgP_{x}y − gfP_{x}y is an absolute zero divisor, and the result will follow since T is nondegenerate.

We have, for all z ∈ T,

P_{[f,g]P_{x}y}z = P_{fgP_{x}y}z + P_{gfP_{x}y}z − {fgP_{x}y, z, gfP_{x}y}.

Now, P_{gfP_{x}y}z = g^{2}f^{2}P_{P_{x}y}z (by ii) = g^{2}f^{2}P_{x}P_{y}P_{x}z = g^{2}P_{f(x)}P_{y}P_{x}z (by ii) = P_{f(x)}g^{2}P_{y}P_{x}z (by i) = f^{2}P_{x}g^{2}P_{y}P_{x}z (by ii) = f^{2}g^{2}P_{x}P_{y}P_{x}z (by i) = f^{2}g^{2}P_{P_{x}y}z = P_{fgP_{x}y}z (by ii).

On the other hand, applying ii and iii, {fgP_{x}y, z, gfP_{x}y} = fg^{2}{P_{x}y, z, fP_{x}y} = f{g^{2}P_{x}y, z, fP_{x}y} = f^{2}{g^{2}P_{x}y, z, P_{x}y} = f^{2}g^{2}{P_{x}y, z, P_{x}y} = 2f^{2}g^{2}P_{P_{x}y}z = 2P_{fgP_{x}y}z.

Therefore, P_{[f,g]P_{x}y}z = P_{fgP_{x}y}z + P_{fgP_{x}y}z − 2P_{fgP_{x}y}z = 0.

1.4 Lemma. Let T be a nondegenerate Jordan triple system, and I an ideal of T. If f, g ∈ Hom_{T} (I, T), then f + g ∈ Hom_{T} (I, T).

Proof: Clearly f +g satisfies properties i and ii, since these are linear conditions on the mapping. Note also that (f + g)P_{I}T ⊆ fP_{I}T + gP_{I}T ⊆ I. Thus it is enough to show that for all x ∈ I and y ∈ T, (f + g)^{2}(P_{x}y) = P_{(f+g)x}y.

We have (f + g)^{2}P_{x}y = f^{2}P_{x}y + g^{2}P_{x}y + gfP_{x}y + fgP_{x}y. Since T is nondegenerate, we can apply Lemma 1.3 to get gfP_{x}y + fgP_{x}y = 2gfP_{x}y = gf(2P_{x}y) = gf{x, y, x} = {f(x), y, g(x)}. Thus (f + g)^{2}P_{x}y = f^{2}P_{x}y + g^{2}P_{x}y + {f(x), y, g(x)} = P_{f(x)}y + P_{g(x)}y + {f(x), y, g(x)} = P_{(f+g)x}y.

1.5 We now turn to the problem of inducing a Φ-module structure on C(T) based on the module structures of the sets Hom_{T} (I, T). The difficulty that we face is that if λ, μ belong to C(T), and (f, I_{f}) ∈ λ, (g, I_{g}) ∈ μ are representatives, it is not clear whether the restrictions of f and g to their common domain I_{f} ∩ I_{g} are again a T -homomorphisms, which would be needed to define λ + μ as the class [f_{I_{f} ∩I_{g}} + g_{I_{f} ∩I_{g}}, I_{f} ∩ I_{g}] (here and in the sequel, h_{L} denotes the restriction of h to L).

Example. Consider the associative commutative algebra Z_{2}[t] of polynomials over Z_{2}. We can form the triple system T = Z_{2}[t]^{(+)} as usual: P_{p}q = pqp. Take the ideal I = Z_{2}[t]t of T, and the map f : I → T given by f(p(t)t) = p(t). Then (f, I) is permissable. Now let L = Z_{2}t^{2} + Z_{2}[t]t^{4}. This is easily seen to be an ideal of T, and it is contained in I. We have f(P_{t^{2}}1) = f(t^{4}) = t^{3} ∉ L, hence f(P_{L}T) ⊈ L, and the restriction of f to L is not a homomorphism.

Let (f, I) be a permissable map of the triple system T and let L be a nonzero ideal of T contained in I. We will say that f restricts to L if f_{L} ∈ Hom_{T} (L, T). It is straightforward that a neccesary and sufficient condition for f to restrict to L is that f(P_{L}T) ⊆ L. Fortunately this holds in a number of cases, as we show in the following lemmas.

1.6 Lemma. Let (f, I_{f}), (g, I_{g})be permissable maps of a nondegenerate Jordan triple system T. Then the set I_{f,g} = {x ∈ I_{f} ∩ I_{g} | f(x) = g(x)} is an ideal of T, and both f and g restrict to I_{f,g}.

Proof: It is clear that if x, y ∈ I_{f,g}, then x + y ∈ I_{f,g}. Also, if z ∈ T, then f(P_{z}x) = P_{z}f(x) = P_{z}g(x) = g(P_{z}x), hence P_{T} I_{f,g} ⊆ I_{f,g}. Similarly {I_{f,g}, T, T } ⊆ I_{f,g}.

Now set v = f(P_{x}z) − g(P_{x}z) with x and z as before. We have P_{v} = P_{f(P_{x}z)} + P_{g(P_{x}z)} − P_{f(P_{x}z),g(P_{x}z)} = f^{2}P_{P_{x}z} + g^{2}P_{P_{x}z} − 2fgP_{P_{x}z} = (f^{2} + g^{2} − 2fg)P_{x}P_{z}P_{x} = P_{f(x)}P_{z}P_{x}+P_{g(x)}P_{z}P_{x}−fg2P_{x}P_{z}P_{x} = P_{f(x)}P_{z}P_{x}+P_{g(x)}P_{z}P_{x}−P_{f(x),g(x)}P_{z}P_{x} (since fg2P_{x} = fgP_{x,x} = P_{f(x),g(x)}) = P_{f(x)−g(x)}P_{z}P_{x} = 0 (since f(x) = g(x)). Therefore v = 0 by nondegeneracy of T, and P_{x}z ∈ I_{f,g}, hence P_{I_{f,g}}T ⊆ I_{f,g}, and I_{f,g} is an ideal of T.

Finally, for all x ∈ I_{f,g} and z ∈ T, f(P_{x}z) = g(P_{x}z) gives gf(P_{x}z) = g^{2}(P_{x}z) = P_{g(x)}z = P_{f(x)}z = f^{2}(P_{x}z) hence f(P_{x}z) ∈ I_{f,g}, and fP_{I_{f,g}}T ⊆ I_{f,g}, and similarly gP_{I_{f,g}}T ⊆ I_{f,g}, hence both f and g restrict to I_{f,g}.

Our next result will allow us to find a common restriction for two permissable maps. Recall that if I, L are ideals of the Jordan triple system T then the set I ∗ L = P_{I}L + P_{T} P_{I}L is again an ideal of T.

1.7 Lemma. Let T be a Jordan triple system and (f, I) be a permissable map. If K, L are ideals of T with K ⊆ I, then f restricts to K ∗ L. In particular, f restricts to K ∗ T.

Proof: Set N = P_{K}P_{L}K + {K ∗ L, T, K ∗ L}. Then we have
P_{K∗L}T ⊆ P_{P_{K}L}T + {P_{K}L, T, P_{T} P_{K}L} + P_{P_{T} P_{K}L}T,

and the containments

P_{P_{K}L}T ⊆ P_{K}P_{L}P_{K}T + {P_{K}L, T, P_{K}L} ⊆ N,

and

P_{P_{T} P_{K}L}T ⊆ P_{T} P_{P_{K}L}P_{T} T + {P_{T} P_{K}L, T, P_{T} P_{K}L} ⊆
P_{T} P_{P_{K}L}T + {K ∗ L, T, K ∗ L} ⊆ N.
Thus, P_{K∗L}T ⊆ N + P_{T} N and f(P_{K∗L}T) ⊆ f(N) + P_{T} f(N).

Now, f(N) = f(P_{K}P_{L}K + {K ∗ L, T, K ∗ L}) = P_{K}P_{L}f(K) + {f(K ∗ L), T, K ∗ L} ⊆ P_{K}P_{L}T + {T, T, K ∗ L} ⊆ K ∗ L. Thus, f(P_{K∗L}T) ⊆ K ∗ L and f restricts to K ∗ L.

Lemma 1.6 gives a better insight into the equivalence relation which defines the extended centroid.

1.8 Lemma. Let T be a nondegenerate Jordan triple system and let (f, I_{f}), (g, I_{g}) be permissable maps. Then the following are equivalent,

(1) (f, I_{f}) ∼ (g, I_{g}),

(2) There is an essential ideal I ⊆ I_{f} ∩ I_{g} such that f_{I} = g_{I} ∈ Hom_{T} (I, T),

(3) f_{I_{f} ∩I_{g}} = g_{I_{f} ∩I_{g}} ∈ Hom_{T} (I_{f} ∩ I_{g}, T).

If in addition T is prime, then these conditions are equivalent to

(4) f(x) = g(x) for some nonzero x ∈ I_{g} ∩ I_{g}.

Proof: (1) ⇒ (2): By Lemma 1.6, I_{f,g} = {x ∈ I_{f} ∩ I_{g} | f(x) = g(x)} is an ideal of T, and since f ∼ g, it contains an essential ideal. Also f_{I_{f,g}} = g_{I_{f,g}} ∈ Hom_{T} (I_{f,g}, T), so we can take I = I_{f,g} in (2).

(2) ⇒ (3): Consider the mapping h = f − g : I_{f} ∩ I_{g} → T. Take x ∈ I_{f} ∩ I_{g} and y ∈ I. Then P_{h(x)}P_{y} = P_{f(x)}P_{y} + P_{g(x)}P_{y} − P_{f(x),g(x)}P_{y} = f^{2}P_{x} + g^{2}P_{x} − 2fgP_{x}P_{y} = P_{x}P_{f(y)} + P_{x}P_{g(y)} − P_{x}P_{f(y),g(y)} = P_{x}P_{f(y)−g(y)} = 0, since f(y) = g(y). Thus the element v = P_{h(x)}y has P_{v} = P_{h(x)}P_{y}P_{h(x)} = 0. Therefore v = 0, since T is nondegenerate, and P_{h(x)}I = 0. This means that h(x) belongs to the annihilator of I (see [20]), but this is an essential ideal, hence h(x) = 0, i.e. f(x) = g(x).

(3) ⇒ (1): This is clear, since if I_{f} and I_{g} are essential, so is I_{f} ∩ I_{g}.

Finally, it is obvious that (4) follows from the above conditions, and, reciprocally, if there is a nonzero x ∈ I_{f} ∩ I_{g} with f(x) = g(x), then, with the notation of 1.6, I_{f,g} is nonzero, hence essential in the prime system T.

1.9 We can now define addition in the extended centroid C(T) of T in the following way: if λ, μ belong to C(T), and (f, I_{f}) ∈ λ, (g, I_{g}) ∈ μ are representatives, then f and g restrict to I = (I_{f} ∩ I_{g}) ∗ T, and we define λ + μ = [f_{I} + g_{I}, I]. The map (f_{I} + g_{I}, I) is permissable by 1.4, 1.7, and the fact that I is essential since both I_{f} and I_{g} are. Moreover, the sum is well defined since if (h, I_{h}) belongs to λ, then f and h agree on I_{f} ∩ I_{h} by 1.8, and, setting I′ = (I_{h} ∩ I_{g}) ∗ T and L = (I_{h} ∩I_{f} ∩I_{g})∗T ⊆ I ∩I′, we have that f_{I}, g_{I}, g_{I′} and h_{I′} restrict to L by Lemma 1.7, and (f_{I} + g_{I})_{L} = f_{L} + g_{L} = h_{L} + g_{L} = (h_{I′} + g_{I′})_{L}. Note also that the ring of scalars Φ acts on C(T) by the rule α[f, I] = [αf, I], and therefore C(T) is a Φ-module.

To define a multiplication in C(T) we need the following fact,

1.10 Lemma. Let T be a nondegenerate Jordan triple system and let I be an ideal of T. There is a bilinear map

Hom_{T} (I, T) × Hom_{T} (I, T) → Hom_{T} (I ∗ T, T)

given by (f, g) ↦ fg : x ↦ f(g(x)), which satisfies fg = gf for all f, g ∈ Hom_{T} (I, T).

Proof: Since g(P_{I}T) ⊆ I, we have g(I ∗ T) ⊆ I, and fg is defined on I ∗ T.

It is obvious that the composition fg defined on I ∗ T satisfies conditions i and iii of the definition of T -homomorphism. We next check condition ii.

Notice first that

P_{I∗T} T ⊆ P_{I}P_{T} P_{I}T + {I ∗ T, T, ∗T } + P_{T} P_{I}P_{T} P_{I}T +

P_{T} {I ∗ T, T, ∗T } ⊆ P_{I}I ∗ T + P_{T} P_{I}I ∗ T + {I ∗ T, T, ∗T } + P_{T} {I ∗ T, T, ∗T }.

Thus

fg(P_{I∗T} T) ⊆ P_{I}fg(I ∗ T) + P_{I}P_{T} fg(I ∗ T) + {fg(I ∗ T), T, ∗T }+

P_{T} {fg(I ∗ T), T, ∗T } ⊆ P_{I}T + P_{T} P_{I}T + {T, T, I ∗ T } + P_{T} {T, T, I ∗ T } ⊆ I ∗ T.

Now we have to show that (fg)^{2}P_{x}z = P_{fg(x)}z for all x ∈ I ∗ T and z ∈ T. It clearly suffices to consider elements x of the form x = P_{y}t with y ∈ I and t ∈ T. For these we have (fg)^{2}(P_{P_{y}t}z) = (fg)^{2}(P_{y}P_{t}P_{y}z) = fg^{2}fP_{y}P_{t}P_{y}z (by Lemma 1.3) = fg^{2}P_{y}P_{t}fP_{y}z) = fP_{g(y)}P_{t}fP_{y}z = f^{2}P_{g(y)}P_{t}P_{y}z = f^{2}g^{2}P_{y}P_{t}P_{y}z = f^{2}g^{2}P_{P_{y}t}z = f^{2}g^{2}P_{x}z = P_{fg(x)}z.

Finally, fg = gf follows from Lemma 1.3.

1.11 We can now define a multiplication in C(T): if λ, μ ∈ C(T), take (f, I_{f}) ∈ λ and (g, I_{g}) ∈ μ. Then (f_{I}, I) and (g_{I}, I), where I = (I_{f} ∩I_{g})∗T, are permissable maps by 1.7. We define the product λμ as the class of (fg_{I∗T} , I∗T), which is permissable by 1.10. This is easily seen to be well defined, and gives C(T) a structure of commutative ring.

1.12 In our construction of the extended centroid C(T) of a nondegenerate triple system T we have not paid attention to the ring of scalars Φ. However, it was implicitely present in the construction since our ideals are Φ-ideals, and our linear maps are Φ-linear. Nevertheless, the resulting ring C(T) is independent of the ring of scalars Φ with which we started. In fact, if we denote by C(T) the extended centroid over the ring Z of integers and by C_{Φ}(T) the extended centroid which arises when we take Φ as the ring of scalars for T (so that, in the second case, we consider Φ-ideals and Φ-module homomorphisms), there is an obvious mapping C_{Φ}(T) → C(T) derived from the fact that every Φ-ideal is a Z-ideal. Now, if λ ∈ C(T) has a representative (f, I) ∈ λ, then I ∗ T is a Φ-ideal (since Φ(I ∗ T) = I ∗ ΦT ⊆ I ∗ T) and we can restrict f to I ∗ T, so that λ always has a representative (f, I), for a Φ-ideal I. Now, if α ∈ Φ, the multiplication by α defines an element α̃ ∈ Hom_{T} (I, T), and we have α̃f = fα̃ by Lemma 1.3. Thus, f is in fact Φ-linear, and the element λ is the image of the class [f, I] in C_{Φ}(T). Therefore the mapping C_{Φ}(T) → C(T) is surjective, and since it is clearly injective, it is a bijection. Moreover, since that mapping is easily seen to preserve sums and products, it is a ring isomorphism.

The basic structural facts about the extended centroid are summarized in the following result.

1.13 Theorem. Let T be a nondegenerate Jordan triple system, then the extended centroid C(T) is a commutative, associative, unital, (von Neumann) regular ring. If, in addition, T is prime, then C(T) is a field.

Proof: It is straightforward that C(T) is a commutative associative ring with the operations defined above. Its unit element is the class of the identity mapping Id ∈ Hom_{T} (T, T).

Now, if (f, I) is permissable, then f(I) is an ideal of T, since P_{I}f(I) = f(P_{I}T) ⊆ f(I) by i, {f(I), T, T } = f({I, T, T }) ⊆ f(I) by iii, and P_{f(I)}T = f^{2}(P_{I}T) ⊆ f(I) by ii.

Thus, we can take an ideal K of T such that K ⊆ I and K ⊕ Kerf is essential in I, and we can assume that f restricts to K (taking K ∗ T if necessary). Then (f, K ⊕ Kerf) is a permissable map wich is equivalent to (f, I). Now, f(K) is an ideal of T, and we can find an ideal L such that L ⊕ f(K) is essential. Define g : L ⊕ f(K) → T by g(a ⊕ f(b)) = b, where a ∈ L and b ∈ K. This is well defined since, if f(b) = 0, then b ∈ K∩Kerf = 0. Moreover, g clearly satisfies properties i and iii of the definition of T -homomorphism. Also g(P_{L⊕f(K)}T) = g(P_{L}T ⊕ P_{f(K)}T) = g(P_{f(K)}T) = gf^{2}(P_{K}T) ⊆ gff(K) ⊆ f(K) ⊆ L ⊕ f(K), and it is easy to see that g also satisfies ii. Therefore (g, L ⊕ f(K)) is permisable.

Now, if c ∈ K, d ∈ Kerf, and d + c belongs to the domain of fgf, we have

fgf(d + c) = fg(f(c)) = f(c) = f(d + c).

This shows that fgf ∼ f, hence for any λ = [f, I] we have found μ = [g, L⊕f(K)] with λμλ = λ. This proves that C(T) is von Neumann regular.

Finally, if T is prime, keeping the previous notation, λ ≠ 0 implies Kerf = 0 by 1.8. Then f(I) is essential, since if N is an ideal of T, then N ∩ f(I) = 0 implies f(N ∗ I) = N ∗ f(I) = 0, hence N ∗ I ⊆ Kerf = 0, and N ⊆ ann_{T} (I) = 0. Thus μ = [g, f(I)] satisfies λμ = μλ = 1, and therefore C(T) is a field.

1.14 Remark. An important property of the elements of the extended centroid is the following “common multiple” property: if λ_{1}, λ_{2}, …, λ_{n} is a finite set of elements of C(T), there are representatives (f_{i}, I) ∈ λ_{i} which share a common domain I. This follows by taking representatives (g_{i}, I_{i}) ∈ λ_{i} and setting I = (⋂_{i=1}^{n} I_{i}) ∗ T. Then each g_{i} restricts to I, and this is essential since the I_{i}’s are. Thus it suffices to put f_{i} = (g_{i})_{I}.

2. Extended centroid of algebras and pairs

In this section we carry out the construction of the extended cetroid for nondegenerate Jordan algebras and pairs, and relate it to the extended centroid of the Jordan triple systems naturally attached to them.

We begin with Jordan algebras. Our first aim is to compare J-homomorphisms of a nondegenerate Jordan algebra J with the those of its underlying triple system. It is clear that that if I is an ideal of J, and f : I → J is a J-homomorphism as a Jordan algebra, then it is a J-homomorphism as a Jordan triple system. We now examine what is needed for the reciprocal.

2.1 Lemma. If J is a nondegenerate Jordan algebra and f : I → J is a Jhomomorphism of the Jordan triple J, then it is a J-homomorphism of Jordan algebras if f(y^{2}) ∈ I for all y ∈ I.

Proof: Take x ∈ T and y ∈ I. We show first that f(x ∘ y) = x ∘ f(y).

By identity QJ 16 of [15], we have

U_{f(y)∘x} = U_{f(y)}U_{x} + U_{x}U_{f(y)} + V_{x}U_{f(y)}V_{x} − U_{U_{x}f(y),f(y)} =
f^{2}(U_{y}U_{x} + U_{x}U_{y} + V_{x}U_{y}V_{x}) − U_{f(U_{y}x),f(y)} (by ii) =
f^{2}(U_{y}U_{x} + U_{x}U_{y} + V_{x}U_{y}V_{x} − U_{U_{y}x,y}) (by iii) =

f^{2}(U_{y∘x}) = U_{f(y∘x)} (by ii).

Next, linearizing QJ 16 of [15], we get

U_{a∘b,c∘b} = U_{b,c}U_{a} + U_{a}U_{b,c} + V_{b}U_{a,c}V_{b} − U_{U_{a,c}b,b},

hence

U_{f(y)∘x,f(y∘x)} = fU_{f(y)∘x,y∘x} = f(U_{f(y),y}U_{x} +U_{x}U_{f(y),y} +V_{x}U_{f(x),y}V_{x} −U_{U_{f(y),y}x,x}) =

f^{2}(U_{y,y}U_{x} + U_{x}U_{y,y} + V_{x}U_{x,y}V_{x} − U_{U_{y,y}x,x}) = 2f^{2}U_{y∘x} = 2U_{f(y∘x)}.

Thus, U_{f(y∘x)−f(y)∘x} = U_{f(y∘x)} + U_{f(y)∘x} − U_{f(y)∘x,f(y∘x)} = U_{f(y∘x)} + U_{f(y∘x)} − 2U_{f(y∘x)} = 0, and, by nondegeneracy of J, f(y ∘ x) = f(y) ∘ x.

Now, U_{f(y)^{2}} = U_{f(y)}^{2} = f^{2}U_{y}f^{2}U_{y} = f^{4}U_{y}^{2} (by i) = f^{4}U_{y^{2}} = U_{f^{2}(y^{2})}.

Also, for all z ∈ J,

{f(y)^{2}, z, f^{2}(y^{2})} = f^{2}{f(y)^{2}, z, y^{2}} = f^{2}(f(y) ∘ {f(y), z, y^{2}}) − f^{2}U_{f(y)}(z ∘ y^{2}) =

f^{2}f^{2}(y ∘ {y, z, y^{2}}) − f^{2}f^{2}U_{y}(z ∘ y^{2}) = f^{4}U_{y^{2},y^{2}}z = U_{f^{2}(y^{2}),f^{2}(y^{2})}z = 2U_{f^{2}(y^{2})}z.

Therefore, U_{f^{2}(y^{2})−f(y)^{2}} = U_{f^{2}(y^{2})}+U_{f(y)^{2}} −U_{f^{2}(y^{2}),f(y)^{2}} = 2U_{f^{2}(y^{2})}−2U_{f^{2}(y^{2})} = 0, and f^{2}(y^{2}) = f(y)^{2} since J is nondegenerate.

2.2 This lemma shows that permissable maps (f, I) of a nondegenerate Jordan algebra J are just permissable maps of the corresponding Jordan triple system that satisfy f(I^{2}) ⊆ I. The equivalence relation that defines the extended centroid of a triple system can be resricted to algebra ideals and algebra J-homomorphisms to obtain the extended centroid of the algebra J, which we temporarily denote by C_{alg}(J). This set can be given a ring structure as for triple systems. Indeed, if (f, g) and (g, I) are permissable maps of the algebra J, then f + g(I^{2}) ⊆ f(I^{2}) + g(I^{2}) ⊆ I, and fg((U_{I}J)^{2}) ⊆ fg(U_{I}U_{J}I^{2} + U_{I}J ∘ U_{I}J) ⊆ fU_{I}U_{J}g(I^{2}) + f(g(U_{I}J) ∘ U_{I}J) ⊆ f(U_{I}J) + f(J ∘ U_{I}J) ⊆ I + J ∘ f(U_{I}J) ⊆ I (so, in fact, if (f, I), (g, I) are permissable for the Jordan triple system J, then fg is permissable for the Jordan algebra J). Therefore we can define addition and multiplication as for triple systems, and the natural mapping C_{alg}(J) → C(J) becomes a homomorphism.

2.3 Proposition. The mapping C_{alg}(J) → C(J) is an isomorphism.

Proof: Let I be an ideal of J as a triple system, and set I_{alg} = {y ∈ I | y^{2} + y ∘ J ⊆ I}. We claim that I_{alg} is an ideal of J as an algebra, and I ∗ J ⊆ I_{alg}.

First, if y, y′ ∈ I_{alg}, then (y + y′) ∘ JI, and (y + y′)^{2} = y^{2} + (y′)^{2} + y ∘ y′ ∈ I, hence y + y′ ∈ I_{alg}.

Next, if z ∈ J and y ∈ I_{alg}, then U_{z}y ∈ I, and (U_{z}y)^{2} = U_{z}U_{y}z^{2} ∈ U_{J}U_{I_{alg}}J ⊆ U_{J}I ∈ I. Now, if x ∈ J, U_{z}y ∘ x = {z, y, z ∘ x} − U_{z}(y ∘ x) ∈ {J, I, J} + U_{J}(I_{alg} ∘ J) ⊆ I + U_{J}I ⊆ I. Therefore U_{J}I_{alg} ⊆ I_{alg}. Also, if y ∈ I_{alg}, then y^{2} ∈ I, y^{4} = U_{y}y ∈ I, and for all x ∈ J, y^{2} ∘ x = {y, y, x} ∈ I, hence I_{alg}^{2} ⊆ I_{alg}.

Now, if z ∈ J, y ∈ I_{alg}, then z ∘ y ∈ I, and (z ∘ y)^{2} = U_{z}y ∘ y − U_{z}y^{2} − U_{y}z^{2} ∈ J ∘ I_{alg} + U_{J}I + U_{I}J ⊆ I, and, if x ∈ J, (z ∘ y) ∘ x = {z, y, x} + {z, x, y} ∈ {J, I, J} + {J, J, I} ⊆ I. Thus I_{alg} ∘ J ⊆ I_{alg}.

Finally, if y ∈ I, z ∈ J, then U_{y}z ∈ I, (U_{y}z)^{2} = U_{y}U_{z}y^{2} ∈ U_{I}J ⊆ I, and for all x ∈ J, U_{y}z ∘ x = {y, z, y ∘ x} − U_{y}(x ∘ z) ∈ {I, J, J} + U_{I}J ⊆ I, hence U_{I}J ⊆ I_{alg}.

Therefore I_{alg} is an ideal of the algebra J, and I ∗ J = U_{I}J + U_{J}U_{I}J ⊆ I_{alg} + U_{J}I_{alg} ⊆ I_{alg}.

Now, if (f, I) is a permissable map of the triple system J, then K = I_{alg} is an essential ideal of the algebra J (since it contains the essential ideal I ∗ J), and f restricts to K ∗ J = U_{K}J. Moreover, for all k ∈ K, and x ∈ J, f((U_{k}x)^{2}) = f(U_{k}U_{x}k^{2}) = U_{k}U_{x}f(k^{2}) (since k^{2} ∈ I) ∈ U_{K}J ⊆ K, and f(U_{K}J ∘U_{K}J) ⊆ f(U_{K}J)∘ U_{K}J ⊆ J ∘ U_{K}J ⊆ U_{K}J. Therefore f((U_{K}J)^{2}) ⊆ U_{K}J, so setting L = U_{K}J, the map (f_{L}, L) is permissable for the algebra J.

This shows that every λ ∈ C(J) has a representative (f, L) ∈ λ with L an ideal of J as an algebra, and f(L^{2}) ⊆ L. According to Lemma 2.1 this means that (f, L) is permissable in the algebra J, therefore λ is the image of [f, L] ∈ C_{alg}(J), and the mapping C_{alg}(J) → C(J) is surjective.

Finally, if λ ∈ C_{alg}(J) is mapped to zero, then for any (f, I) ∈ λ, f = 0 by Lemma 1.8. Thus the mapping is injective, and it is an isomorphism.

In view of this result we can identify the extended centroids of an algebra J and of its underlying triple system, and denote it simply by C(J).

2.4 We now consider Jordan pairs. These are related to Jordan triple systems through their attached polarized systems. Also, if V = (V^{+}, V^{−}) is a Jordan pair, and I = (I^{+}, I^{−}) is an ideal of V, then the set T (I) = I^{+} ⊕I^{−} is, up to a natural isomorphism, an ideal of T (V) = V^{+} ⊕ V^{−}. Thus for any f = (f^{+}, f^{−}) ∈ Hom_{V} (I, V), we obtain a mapping t(f) = f^{+} ⊕ f^{−} : T (I) → T (V), given by t(f)(x^{+} ⊕ x^{−}) = f^{+}(x^{+}) ⊕ f^{−}(x^{−}), which is easily seen to belong to Hom_{T (V)}(T (I), T (V)).

The lemmas of section 1 have then a direct translation to V, and we can form the extended centroid of a nondegenerate V as the set of equivalence classes of permissable maps of V under the equivalence relation “∼” given by: (f, I) ∼ (g, L) if there is an essential ideal K if V with K^{σ} ⊆ I^{σ} ∩ L^{σ}, and f^{σ}(x) = g^{σ}(x) if x ∈ K^{σ}, for σ = ±. The quotient set, which we denote C(V), has a structure of commutative ring, and will be called the extended centroid of V.

Notice that since every permissable map ((f^{+}, f^{−}), (I^{+}, I^{−})) of V gives rise to a permissable map (f^{+} ⊕ f^{−}, I^{+} ⊕ I^{−}) of T (V), it determines an element of C(T (V)). In fact, this defines a homomorphism of C(V) into C(T (V)) due to the following:

2.5 Lemma. If f, g are permissable maps of the Jordan pair V, then f ∼ g if and only if t(f) ∼ t(g).

Proof: If t(f) and t(g) agree on the essential ideal K of T (V), then they agree on the polarized ideal T (V) ∗ K = (P_{V^{+}}K + P_{V^{+}}P_{V^{−}}K) ⊕ (P_{V^{−}}K + P_{V^{−}}P_{V^{+}}K), which is also essential. Then f and g agree on ((T (V) ∗ K)^{+}, (T (V) ∗ K)^{−}), hence f ∼ g. The reciprocal is clear.

Thus, we have a monomorphism C(V) → C(T (V)).

2.6 Proposition. The mapping C(V) → C(T (V)) is an isomorphism.

Proof: We only have to show that the mapping is surjective. For that, if λ ∈ C(T (V)) and (f, I) ∈ λ, we can restrict f to the polarized ideal (T (V) ∗ I) ∗ T (V) by Lemma 1.7, so we can assume that I is polarized, I = I^{+} ⊕ I^{−}, with I^{σ} ⊆ V^{σ}. Thus, to show that λ comes from an element of C(V), it suffices to show that f is polarized, f(I^{σ}) ⊆ V^{σ}, σ = ±, since in this case we will have f = f^{+} ⊕ f^{−} = t(f^{+}, f^{−}), where f^{σ} = f_{I^{σ}} is the corresponding restriction.

Now, if x^{σ} ∈ I^{σ}, put f^{σ} = y^{+} ⊕ y^{−}. Then 0 = P_{x^{σ}}V^{σ}, hence 0 = f^{2}(P_{x^{σ}}V^{σ}) = P_{f(x^{σ})}V^{σ} = Q_{y^{+}⊕y^{−}}V^{σ} = Q_{y^{−σ}}V^{−σ} = Q_{y^{−σ}}T (V). Thus y^{−σ} = 0 by nondegeneracy, and f(x^{σ}) = y^{σ} ∈ V^{σ}.

3. Extended central closure

3.1 In this section we define the scalar extension of a Jordan system which is naturally associated to its extended centroid. We carry out the construction for a nondegenerate Jordan triple system T, but we point out that the results are also valid for algebras and pairs with the suitable modifications in the latter case, which we mention at the end of this section.

First form the free scalar extension C(T) ⊗ T. Since, at the end, the result will be independent of the ring of scalars Φ, we will not worry about subscripts of the tensor product, so it can be taken over any ring of scalars Φ for T.

We consider the set

R = {∑_{i}(ρ_{i}λ_{i} ⊗ x_{i} − ρ_{i} ⊗ f_{i}(x_{i})) | λ_{i}, ρ_{i} ∈ C(T), (f_{i}, I_{i}) ∈ λ_{i}, and x_{i} ∈ I_{i}}.

3.2 Lemma. R is an ideal of C(T) ⊗ T.

Proof: For any multiplication operator M ∈ M(T), and any u = ρλ ⊗ x − ρ ⊗ f(x) ∈ R we have 1 ⊗ M(u) = ρλ ⊗ Mx − ρ ⊗ f(Mx) ∈ R, using the obvious identification of multiplication algebras M(C(T) ⊗ T) = C(T) ⊗ M(T). Thus R is an outer ideal.

Now, if u = ρλ ⊗ x − ρ ⊗ f(x) ∈ R, and μ ⊗ z ∈ C(T) ⊗ T, we have P_{u}μ ⊗ z = P_{ρλ⊗x}(μ ⊗ z) + P_{ρ⊗f(x)}(μ ⊗ z) − P_{ρλ⊗x,ρ⊗f(x)}(μ ⊗ z) = μρ^{2}λ^{2} ⊗ P_{x}z + μρ^{2} ⊗ P_{f(x)} − μρ^{2}λ ⊗ P_{x,f(x)}z = μρ^{2}λ^{2} ⊗ P_{x}z + μρ^{2} ⊗ f^{2}P_{x} − 2μρ^{2}λ ⊗ fP_{x}z = ((μρ^{2}λ)λ ⊗ P_{x}z − μρ^{2}λ ⊗ f(P_{x}z)) − ((μρ^{2})λ ⊗ f(P_{x}z) − μρ^{2} ⊗ f^{2}(P_{x}z)) ∈ R. Thus R is an ideal of C(T) ⊗ T.

3.3 The first step in the construction of the scalar extension associated to the extended centroid is to form the quotient T̃ = C(T)⊗T/R. This is a triple system over the ring C(T) (since the ideal R is clearly C(T)-invariant), and the natural injection T → C(T) ⊗ T, a ↦ 1 ⊗ a, induces a homomorphism T → T̃. To show that this is a scalar extension we must show that this is also an injection, that is, (1 ⊗ T) ∩ R = 0. We follow the ideas of [12, 9] with the changes needed to deal with the nonlinearity of the multiplication in T.

Let I be an essential ideal of T. We denote by (C(T) ⊗ T)_{I} the set of elements a ∈ C(T) ⊗ T that can be written in the form a = ∑ λ_{i} ⊗ x_{i} for some elements λ_{i} ∈ C(T) which admit representatives (f_{i}, I) ∈ λ_{i} whose domain is the ideal I. For each y ∈ I we want to define a map F_{y} : (C(T) ⊗ T)_{I} → T by F_{y}(a) = ∑ f_{i}(P_{y}x_{i}) where a = ∑ λ_{i} ⊗ x_{i}, and (f_{i}, I) ∈ λ_{i} as before. Note that since y ∈ I, P_{y}x_{i} belongs to I, hence to the domain of the f_{i}’s.

3.4 Lemma. The mappings F_{y} are well defined.

Proof: We must show that if ∑ λ_{i} ⊗ x_{i} = 0, where each λ_{i} ∈ C(T) has a representative (f_{i}, I) ∈ C(T), then ∑ f_{i}(P_{y}x_{i}) = 0. We can assume that the elements x_{i} are part of a system of generators of the Φ-module T. Then, by (8.8) of [27], there are elements α_{ij} ∈ Φ, and a finite number of elements μ_{j} ∈ C(T) such that ∑_{i} x_{i}α_{ij} = 0 for every j, and λ_{i} = ∑_{j} α_{ij}μ_{j} for every i.

We can take an essential ideal L of T, contained in I, and representatives (g_{j}, L) ∈ μ_{j}. Then, for all z ∈ L ∗ T, we have P_{∑_{i} f_{i}(P_{y}x_{i})}z = ∑_{i} P_{f(P_{y}x_{i})}z + ∑_{i<k} P_{f_{i}(P_{y}x_{i}),f_{k}(P_{y}x_{k})}z. Now, P_{f_{i}(P_{y}x_{i})}z = f_{i}^{2}(P_{P_{y}x_{i}}z) = (∑_{j} α_{ij}g_{j})^{2}(P_{P_{y}x_{i}}z) (since P_{P_{y}x_{i}}z ∈ L∗T is in the domain of all products g_{j}g_{l}, and, there, f_{i} = ∑_{j} α_{ij}g_{j} by 1.8). = ∑_{j} P_{P_{y}α_{ij}x_{i}}g_{j}^{2}(z) + ∑_{j<l} P_{P_{y}α_{ij}x_{i},P_{y}α_{il}x_{i}}g_{j}g_{l}(z). Similarly, P_{f_{i}(P_{y}x_{i}),f_{k}(P_{y}x_{k})}z = ∑_{j,l} P_{P_{y}α_{ij}x_{i},P_{y}α_{kl}x_{k}}g_{j}g_{l}(z). Thus,

P_{f_{i}(P_{y}x_{i})}z = ∑_{i}(∑_{j} P_{P_{y}α_{ij}x_{i}}g_{j}^{2}(z)+

∑_{j<l} P_{P_{y}α_{ij}x_{i},P_{y}α_{il}x_{i}}g_{j}g_{l}(z)) + ∑_{i<k} ∑_{j,l} P_{P_{y}α_{ij}x_{i},P_{y}α_{kl}x_{k}}g_{j}g_{l}(z) =

∑_{j}(∑_{i} P_{P_{y}α_{ij}x_{i}}g_{j}^{2}(z) + ∑_{i<k} P_{P_{y}α_{ij}x_{i},P_{y}α_{kj}x_{k}}g_{j}^{2}(z))+

∑_{j<l}(∑_{i} P_{P_{y}α_{ij}x_{i},P_{y}α_{il}x_{i}}g_{j}g_{l}(z) + ∑_{i<k} P_{P_{y}α_{ij}x_{i},P_{y}α_{kl}x_{k}}g_{j}g_{l}(z)) =

∑_{j} P_{P_{y}(∑_{i} α_{ij}x_{i})}g_{j}^{2}(z) + ∑_{j<l} P_{P_{y}(∑_{i} α_{ij}x_{i}),P_{y}(∑_{k} α_{kl}x_{k})}g_{j}g_{l}(z) = 0.

Therefore ∑_{i} f(P_{y}x_{i}) ∈ ann_{T} (L ∗ T) by 1.7 of [20], and ∑_{i} f_{i}(P_{y}x_{i}) = 0 since L ∗ T is essential.

3.5 Lemma. The mapping T → T̃ = C(T) ⊗ T/R is injective.

Proof: Suppose that 1 ⊗ x ∈ 1 ⊗ T ∩ R. Then there are λ_{i}, ρ_{i} ∈ C(T) and x_{i} ∈ T such that 1 ⊗ x = ∑(ρ_{i}λ_{i} ⊗ x_{i} − ρ_{i} ⊗ f_{i}(x_{i})), where (f_{i}, I) ∈ λ_{i} and x_{i} ∈ I. We can find an essential ideal L of T such that all products τ_{1}τ_{2} with τ_{k} = 1, ρ_{i} or λ_{i} have representatives whose common domain is L. Then, for all y ∈ L we have F_{y}(ρ_{i}λ_{i} ⊗ x_{i} − ρ_{i} ⊗ f_{i}(x_{i})) = f_{i}g_{i}(P_{y}x_{i}) − g_{i}P_{y}f_{i}(x_{i}) = f_{i}g_{i}(P_{y}x_{i}) − f_{i}g_{i}(P_{y}x_{i}) = 0, where (g_{i}, L) ∈ ρ_{i}. Thus F_{y}(1 ⊗ x) = 0.

On the other hand, F_{y}(1⊗x) = P_{y}x, hence we obtain P_{L}x = 0, and x ∈ ann_{T} (L) (by (5.4)(b) of [21]). Therefore x = 0, and 1 ⊗ x = 0.

3.6 According to the previous lemma, we can identify T with its image in T̃. Now, for any ideal I of T, C(T) ⊗ I is an ideal of C(T) ⊗ T, and Ĩ = (C(T) ⊗ I + R)/R is an ideal of T̃. If I_{1} ⊆ I_{2} are ideals of T then annT̃(Ĩ_{1}) ⊇ annT̃(Ĩ_{2}), and therefore the set M̃ = ∪{annT̃(Ĩ) | I is an essential ideal of T }, is an ideal of T̃. Moreover, since annT̃(Ĩ) ∩ T ⊆ ann_{T} (I) = 0 for any essential ideal I of T, we have M̃ ∩ T = 0. The system C(T)T = T̃/M̃ will be called the extended central closure of T . Since T ∩ M̃ = 0, we have T ⊆ C(T)T, and C(T)T is a scalar extension of T.

3.7 Lemma. Any ideal Ñ of T̃ with Ñ ∩ T = 0 is contained in M̃.

Proof: Take an ideal N ⊇ R of C(T) ⊗ T with Ñ = N/R. Any a ∈ N has the form a = ∑ λ_{i} ⊗ x_{i}, with x_{i} ∈ T, λ ∈ C(T), and we can take an ideal I of T such that all λ_{i} and λ_{i}λ_{j} have representatives (f_{i}, I) and (f_{i}f_{j}, I) respectively, whose domain is I. Now, writing ≡ for the congruence mod R, for any y ∈ I we have P_{1⊗y}a ≡ 1 ⊗ (∑_{i} f_{i}(P_{y}x_{i})). Thus, 1 ⊗ ∑_{i} f_{i}(P_{y}x_{i}) ∈ N + R = N, and it also belongs to 1 ⊗ T. Since T ∩ N = 0, we get P_{1⊗y}a = 0.

Similarly, we get P_{1⊗y,1⊗y′}a = {a, 1 ⊗ y, 1 ⊗ z} = {a, 1 ⊗ z, 1 ⊗ y} = {1 ⊗ y, a, 1 ⊗ z} = 0 for all y, y′ ∈ I, and all z ∈ T. Also, since λ_{i}λ_{j} admits a representative f_{i}f_{j} whose domain is I, we obtain in the same way P_{a}(1 ⊗ y) = P_{1⊗y}P_{a}(1 ⊗ z) = P_{1⊗y,1⊗y′}P_{a}(1 ⊗ z) = P_{a}P_{1⊗y}(1 ⊗ z) = P_{a}P_{1⊗y,1⊗y′}(1 ⊗ z) = 0.

These equalities imply PĨã = PĨPãT̃ = PãPĨT̃ = PãĨ = {Ĩ, ã, T̃} = {ã, Ĩ, T̃} = {ã, T̃, Ĩ} = 0, where ã = a + R ∈ T̃. Therefore ã ∈ annT̃(Ĩ) ⊆ M̃, and Ñ ⊆ M̃.

3.8 Theorem. The extended central closure C(T)T of a nondegenerate Jordan system T is a tight scalar extension of T. Therefore it is nondegenerate, and if T is strongly prime, then C(T)T is strongly prime.

Proof: T ⊆ C(T)T is tight by Lemma 3.7. The other assertions follow from this and [22].

3.9 To close this section we make some remarks on the construction of the extended central closure for algebras and pairs.

If J is a nondegenerate Jordan algebra, the previous construction can be carried out in exactly the same way, and it produces a Jordan algebra. To see that, it is enough to notice that all the ideals involved in the construction (R and M̃) are algebra ideals. This follows by allowing z = 1 in the proof of 3.2 and making use of conditions iv and v of the definition of J-homomorphism. The resulting algebra C(J)J will be called the extended central closure of J. Note that, by 2.3, the extended central closure of J is the same whether we consider J as an algebra or a triple system.

As for nondegenerate Jordan pairs V = (V^{+}, V^{−}), the construction can be carried out with the obvious modifications. We start from the free scalar extension C(V) ⊗ V = (C(V) ⊗ V^{+}, C(V) ⊗ V^{−}) and consider the pair-ideal R = (R^{+}, R^{−}), where R^{σ} is the linear span of the elements of the form ρλ ⊗ x^{σ} − ρ ⊗ f^{σ}(x^{σ}) where λ has a representative ((f^{+}, f^{−}), (I^{+}, I^{−})), and x^{σ} ∈ I^{σ}. The rest of the construction is the same as for triple systems, using the pair version of the corresponding notions, and it produces a Jordan pair which we denote by C(V)V = (C(V)V^{+}, C(V)V^{−}), and call the extended central closure of V. Notice that there is an obvious identification T (C(V)V) = C(T (V))V.

In both cases (algebras and pairs) the extended central closure is a tight scalar extension of the original Jordan system, hence it is nondegenerate and inherits strong primeness from the original system.

4. Properties of the extended central closure

4.1 As for associative algebras, the extended central closure of a Jordan system should be considered as a localization of the algebra (cf. [28]). To reflect this we introduce the following notions.

Let V ⊆ Ṽ be Jordan pairs, let I be an ideal of V, and let a ∈ Ṽ^{σ}, σ = ±. An element x ∈ V^{σ} will be called a V -denominator of a into I if the elements Q_{x}a, Q_{a}x, and the sets Q_{x}Q_{a}V^{−σ}, Q_{a}Q_{x}V^{σ}, D_{x,a}V^{−σ} and D_{a,x}V^{σ} are all contained in I. We denote by D_{V} (a, I) the set of denominators of a into I. If I = V, we simply write D_{V} (a).

For Jordan triple systems T ⊆ T̃, a ∈ T̃ and I an ideal of T we define the T -denominators of a into I similarly, by deleting the superscripts ± in the previous definition. The resulting set will be denoted D_{T} (a, I), or simply D_{T} (a) if T = I.

Finally, for Jordan algebras J ⊆ J̃, a ∈ J̃ and an ideal I of J, we add the conditions U_{x}U_{a}Ĵ+ U_{a}U_{x}Ĵ+ {a, x, Ĵ} ⊆ I, where Ĵ is a unital hull of J. Again, the set of J-denominators of a into I will be denoted by D_{J}(a, I), or D_{J}(a) if I = J.

4.2 Lemma. Let V ⊆ Ṽ be Jordan pairs, I be an ideal of V and a ∈ V^{σ}, σ = ±. Then D_{V} (a, I) is an inner ideal of V.

Proof: If x, y ∈ D_{V} (a, I) ⊆ V^{−σ}, D_{x+y,a}V^{−σ} ⊆ I^{−σ}, and Q_{a}(x + y), and D_{a,x+y}V^{σ} are contained in I^{σ} by linearity.

Next, Q_{x,y}Q_{a} = D_{x,a}D_{y,a} − D_{x,Q_{a}y} by JP9 of [16]. Thus, Q_{x,y}Q_{a}V^{−σ} ⊆ I^{−σ}, hence Q_{x+y}Q_{a}V^{−σ} ⊆ Q_{x,y}Q_{a}V^{−σ} + Q_{x}Q_{a}V^{−σ} + Q_{y}Q_{a}V^{−σ} ⊆ I^{−σ}, and Q_{x+y}a = Q_{x}a + Q_{y}a + Q_{x,y}a ∈ I^{−σ}

Now, by JP6 of [16],

Q_{a}Q_{x,y} = D_{a,y}D_{a,x} + D_{Q_{a}x,y} − D_{a,Q_{x,y}a},

and

D_{a,Q_{x,y}a} = D_{Q_{a}x,y} + D_{Q_{a}y,x}

by linearized JP2 of [16]. Thus, Q_{a}Q_{x,y} = D_{a,y}D_{a,x} − D_{Q_{a}y,x}, and Q_{a}Q_{x,y}V^{σ} ⊆ I^{σ}. Therefore Q_{a}Q_{x+y}V^{σ} ⊆ I^{σ}, and we finally get x + y ∈ D_{V} (a, I).

Now take z ∈ V^{σ}. Then, for any x ∈ D_{V} (a, I), we have Q_{a}Q_{x}z ∈ I^{σ} and Q_{Q_{x}z}a = Q_{x}Q_{z}Q_{x}a ∈ Q_{x}Q_{z}I^{−σ} ⊆ I^{−σ}.

We also have the contaiments:

Q_{a}Q_{Q_{x}z}V^{σ} ⊆ Q_{a}Q_{x}V^{σ} ⊆ I^{σ},
Q_{Q_{x}z}Q_{a}V^{−σ} = Q_{x}Q_{z}Q_{x}Q_{a}V^{−σ} ⊆ Q_{x}Q_{z}I^{−σ} ⊆ I^{−σ},

D_{Q_{x}z,a}V^{−σ} ⊆ D_{x,D_{a,x}z}V^{−σ} + D_{Q_{x}a,z}V^{−σ}(by linearized JP2 of [16]) ⊆

D_{x,I^{σ}}V^{−σ} + D_{I^{σ},z}V^{−σ} ⊆ I^{−σ},

and

D_{a,Q_{x}z}V^{σ} ⊆ D_{z,Q_{x}a}V^{σ} + D_{x,D_{a,x}z}V^{σ} (by linearized JP2 of [16]) ⊆

D_{z,I^{−σ}}V^{σ} + D_{x,I^{−σ}}V^{σ} ⊆ I^{σ}.

Thus Q_{x}z ∈ D_{V} (a, I).

Similarly, we obtain that denominator sets are inner ideals in Jordan triple systems and algebras.

The proximity between a Jordan system and its extended central closure can be expressed in terms of denominators.

4.3 Lemma. Let T be a nondegenerate Jordan system (triple, pair or algebra). Then

(1) Every nonzero inner ideal of C(T)T has nonzero intersection with T.

(2) For every a ∈ C(T)T, and every essential ideal I of T, D_{T} (a, I) contains an essential ideal of T,

Proof: We first prove (2). Write a = ∑_{i=1}^{n} λ_{i}x_{i} with λ_{i} ∈ C(T), and x_{i} ∈ T. For each i we take a representative (f_{i}, I_{0}) ∈ λ_{i}. Set L = I_{0} ∩ I. Then L ∗ L ⊆ I_{0}, and f_{i} restricts to L ∗ L by Lemma 1.7, for all i. Now, using that the elements λ_{i} act as the corresponding T -homomorphisms f_{i} on the elements of L ∗ L and L ∗ T, that f_{i}(L ∗ L) = L ∗ f_{i}(L) ⊆ L ∗ T ⊆ L ⊆ I, and that f_{i}f_{j}(L ∗ L) ⊆ f_{i}(L ∗ T) ⊆ L ⊆ I, a direct computation shows that L ∗ L ⊆ D_{T} (A, I). Moreover, since I and I_{0} are essential ideals, L is essential, hence L ∗ L is essential.

Now, to prove (1), let K be a nonzero inner ideal of C(T)T, and take a ∈ K. By (1) there is an essential ideal I ⊆ D_{T} (a). By [20], P_{a}I = 0 implies that a ∈ ann_{C(T)}(C(T)I) since C(T)T is nondegenerate and C(T)I is an ideal of C(T)T. Then 0 ≠ ann_{C(T)T} (C(T)I) ∩ T ⊆ ann_{T} (I) by tightness, and this is a contradiction since I is essential. Thus K ∩ T ⊇ P_{a}I is nonzero.

The key fact about the extended centroid of strongly prime systems is that it captures scalars in the centroid of any extension T ⊆ T̃ as soon as they do not remove some nonzero ideal from T: if λ ∈ Γ(T̃) has λI ⊆ T for some nonzero ideal I of T, then λ ∈ C(T). The argument is essentially the same as in Lemmas 5.1 and 5.2 of [2]. In fact we will show next that inner ideals are enough due to the following result (see Lemma 5.1 of [2]). Recall that a structural transformation (f, g) : T ⇀ ↽ T′ is pair of mappings f : T → T′ and g : T′ → T satisfying P_{f(x)} = fP_{x}g for all x ∈ T, and P_{g(y)} = gP_{y}f for all y ∈ T′. (We will also say that f is structural if there is g such that (f, g) is structural.)

4.4 Lemma. Let T be a Jordan system and let M_{1}(T) be its (unital) multiplication algebra (spanned by the identity on T and all the operators P_{x}, D_{x,y} for x, y ∈ T –toghether with P_{x,1} if T is an algebra). If K is an inner ideal of T, then M_{1}(T)K is an ideal of T.

Proof: Clearly M_{1}(T)K is outer, hence it suffices to show that this is an inner ideal.

Notice that if x, y ∈ T, L_{x,y} = Id + P_{x}P_{y} − B_{x,y} is a sum of structural transformation, and the composition of structural transformations is again structural. Thus M_{1}(T) is generated an a Φ-module by structural transformations. Then, any element from M_{1}(T)K has the form z = ∑_{i} f_{i}k_{i} for structural transformations f_{i} = Id, P_{x_{i}} or B_{x_{i},y_{i}}, or compositions of these, and elements k_{i} from K. (We write g_{i} for the second part of the structural transformation (f_{i}, g_{i}).)

Now, we have P_{f_{i}k_{i}}T = f_{i}P_{k_{i}}g_{i}T (by structurality of f_{i}) ⊆ f_{i}K (by innerness of K) ⊆ M_{1}(T)K, and P_{f_{i}k_{i},f_{j}k_{j}}T ⊆ L_{f_{i}k_{i},T} f_{j}k_{j} ⊆ M_{1}K, yielding innerness of M_{1}K.

4.5 Proposition. Let T ⊆ T̃ be Jordan systems. Suppose that T is strongly prime, and let λ ∈ Γ(T̃). Then there is a permissable map (f, I) of T such that f(x) = λx for all x ∈ I if and only if there is a nonzero inner ideal K of T such that λK ⊆ T.

Moreover, this always happens for λ = μ^{2} if there is a nonzero x ∈ T with μx ∈ T, i.e. if T ∩ μT ≠ 0.

Proof: The “only if” part is clear. Now, if λK ⊆ T for some nonzero inner ideal K of T, then λM_{1}(T)K = M_{1}λK ⊆ T. So, denoting by L the ideal M_{1}(T)K, we have λL ⊆ T. This is a nonzero ideal (since it contains K) hence it is essential by primeness of T.

It is easy to see that P_{L∗T} T ⊆ P_{L}P_{T} L + P_{T} P_{L}P_{T} L + {L, T, L}, and from this, that λP_{L∗T} T ⊆ L∗T. Thus, setting I = L∗T, we can consider the mapping f : I → T given by f(x) = λx, which is permissable. This proves the first assertion.

Finally, if μ ∈ Γ(T) has some nonzero μx ∈ T ∩ μT, then μ^{2}P_{x}T = P_{μx}T ⊆ T, and the previous part applies to λ = μ^{2}.

As a first consequence we can relate the extended centroid C(T) of a strongly prime Jordan system T and the extended centroid C(C(T)T) of its central closure. Note that since the centroid Γ(C(T)T) contains C(T), we can consider C(T) contained in C(C(T)T).

4.6 Lemma. Let J be a strongly prime Jordan system, then C(C(T)T)^{2} ⊆ C(T).

Proof: Take λ ∈ C(C(T)T), and a representative (f, I) ∈ λ. Since T ⊆ C(T)T is tight, we can take a nonzero element y ∈ I ∩ T. Then, by there is an essential ideal L of T contained in D_{T} (f(y)), and the inner ideal P_{y}L of T is nonzero since L is essential (see 1.7 of [20]). Now we have f^{2}(P_{y}L) = P_{f(x)}z ⊆ T, i.e., λ^{2}P_{y}L ⊆ T. Thus it suffices to apply 4.5.

4.7 In what remains of this section we examine to what extent the extended central closure is determined by the properties 4.3. Extended central closures of linear algebras can be seen under the light of the notions of localization theory, through the category of modules over the corresponding multiplication algebras, and some of its subcategories (see [28]). In our case, however, multiplication algebras do not seem to be the adequate tools for studying extended central closures since they do not reflect the nonlinearity of multiplications in Jordan systems. Nevertheless, our next results have some “injective flavour”, since they deal with liftings of homomorphisms. Consequently, they can be considered as a very preliminar step towards a categorical aproach to the extended central closure, in which the use of modules over the multiplication algebra M_{1}(T) of T is substituted by extensions of the Jordan system. These are, of course, modules for M_{1}(T), but they also take into acount the nonlinear action y ↦ P_{y}x.

Let T be a nondegenerate Jordan system over Φ, and let T̃ be an extension of T over the Φ-algebra Ω. We denote by Ω ∩ C(T) the set {ω ∈ Ω | there is a permissable map (f, I) of T such that f(y) = ωy for all y ∈ I}.

If ω ∈ Ω ∩ C(T), and (f, I), (g, L) are permissable maps of T with f(x) = ωx for all x ∈ I, and g(y) = ωy for all y ∈ L, then it is clear that (f, I) and (g, L) are equivalent, hence they define a unique element σ(ω) of C(T). In this way we get a mapping σ : Ω ∩ C(T) → C(T). Moreover, it is easy to check that this map σ is a homomorphism of rings.

4.8 Lemma. Let T ⊆ T̃ be an extension of Jordan Φ-systems with T nondegenerate, and suppose that T̃ is a system over the Φ-algebra Ω. Then there is a homomorphism ϕ : (Ω ∩ C(T))T → C(T)T given by ϕ(∑ ω_{i}x_{i}) = ∑ σ(ω_{i})x_{i}.

Proof: To see that ϕ is well-defined, soppose that ∑ ω_{i}x_{i} = 0, where ω_{i} ∈ Ω and x_{i} ∈ T ⊆ T̃. We must show that z = ∑ σ(ω_{i})x_{i} = 0.

Take an essential ideal I of T, and representatives (f_{i}, I) ∈ σ(ω_{i}), such that for all i, j, f_{i}f_{j} is defined on I. Then, for all y ∈ I, P_{z}y = P_{∑ σ(ω_{i})x_{i}}y = ∑ P_{σ(ω_{i})x_{i}}y + ∑_{i<j} P_{σ(ω_{i})x_{i},σ(ω_{j})x_{j}}y = ∑ σ(ω_{i})^{2}P_{x_{i}} + ∑_{i<j} σ(ω_{i})σ(ω_{j})P_{x_{i},x_{j}}y = ∑ f_{i}^{2}(P_{x_{i}}y) + ∑_{i<j} f_{i}f_{j}(P_{x_{i},x_{j}}y) = ∑ ω_{i}^{2}(P_{x_{i}}y) + ∑_{i<j} ω_{i}ω_{j}(P_{x_{i},x_{j}}y) = P_{∑ ω_{i}x_{i}}y = 0. Thus P_{z}y = 0 for all y ∈ I, hence P_{z}C(T)T = 0, and z ∈ ann_{C(T)T} (C(T)I) = 0.

4.9 Proposition. Let T ⊆ T̃ be a tight extension of strongly prime Jordan systems. If any inner ideal of T̃ meets T, and for any x ∈ T̃, D_{T} (x) contains an essential ideal of T, then there is a unique homomorphism ϕ : Γ(T̃)^{2}T (⊆ T̃) → C(T)T which is the identity on T. Moreover, ϕ is a monomorphism.

Proof: Take λ ∈ Γ(T̃). For any nonzero x ∈ T, there is a nonzero ideal I of T contained in D_{T} (λx). Then P_{x}I is a nonzero inner ideal of T, and λ^{2}P_{x}I = P_{λx}I ⊆ T. Therefore λ^{2} ∈ Γ(T̃) ∩ C(T) by 4.5, and we wet Γ(T̃)^{2} ⊆ Γ(T̃) ∩ C(T). Then the restiction of the homomorphism of Lemma 4.8 gives a homomorphism ϕ : Γ(T̃)^{2}T → C(T)T, and this is clearly the identity on T.

Now, if ψ : Γ(T̃)^{2}T → C(T) is a homomorphism with ψ_{T} =Id, take λ ∈ Γ(T̃)^{2}, and x ∈ T. There is a nonzero ideal I of T with λI + λ^{2}I ⊆ T (since both λ and λ^{2} are represented by T -homomorphisms). Then, for all y ∈ I we haveP_{ϕ(λx)−ψ(λx)}y = P_{ϕ(λx)}y + P_{ψ(λx)}y − P_{ϕ(λx),ψ(λx)}y = P_{ϕ(λx)}ϕ(y) + P_{ψ(λx)}ψ(y) − P_{ϕ(λx),ϕ(ψ(λx))}ϕ(y) (since ϕ and ψ are the identity on T) = ϕP_{λx}y + ψP_{λx}y − ϕP_{λx,ψ(λx)}y = ϕ(P_{x}λ^{2}y) + ψ(P_{x}λ^{2}y) − ϕ(P_{x,ψ(λx)}λy) = 2P_{x}λ^{2}y − P_{x,λx}λy = 2P_{x}λ^{2}y − P_{ψ(x),ψ(λx)}ψ(λy) (since ψ is the identity on T) = 2P_{x}λ^{2}y − ψ(P_{x,λx}λy) = 2P_{x}λ^{2}y − ψ(2P_{x}λ^{2}y) = 2P_{x}λ^{2}y − 2P_{x}λ^{2}y = 0.

Thus ϕ(λx) − ψ(λx) ∈ ann_{C(T)T} (C(T)I) = 0, hence ϕ(λx) = ψ(λx) for all λ ∈ Γ(T)^{2}, and all x ∈ T. Therefore ϕ = ψ.

Finally, let us show that ϕ is a monomorphism. Indeed, if x̃ = ∑ λ_{i}x_{i}, with λ_{i} ∈ Γ(T̃)^{2} and x_{i} ∈ T, has ϕ(x̃) = 0, then take an essential ideal I of T such that all the elements σ(λ_{i}) and σ(λ_{i})σ(λ_{j}), where σ is the homomorphism of 4.7, have representatives defined on I. Then it is easy to see that x̃ ∈ annT̃(I). But this is an inner ideal, and satisfies annT̃(I) ∩ T ⊆ ann_{T} (I) = 0. Therefore annT̃(I) = 0, and x̃ = 0.

4.10 When the characteristic is two, the extended central closure C(T)T of a Jordan system T, as we have defined it, may not really be a closure in the sense that it is for linear algebras, since we may not have Γ(C(T)T) = C(T), or even C(C(T)T) = C(T). The problem here is the same as in the construction [23] of a (ordinary) central closure of a Jordan algebra, that is, a central extension J̄ of a strongly prime Jordan algebra J whose centroid is the field of fractions of the centroid of J. It is not difficult, however, to obtain a true closure of a Jordan system following the ideas of [23]. To do that one can iterate the construction of the extended centroid and the extended central closure of T by transfinite induction: start from a strongly prime system T, and set C_{1}(T) = C(T), then define C_{β}(T) = C(C_{α}(T)T) and C_{β}(T)T = C(C_{α}(T)T)(C_{α}(T)T) if β = α + 1 is not a limit ordinal, and C_{β}(T) = ∪_{α<β}C_{α}(T), C_{β}(T)T = ∪_{α<β}C_{α}(T)T if β is a limit. By 4.6, at each step we get an extension of exponent two of the previous extended centroid, so we can take all them inside the perfect closure of C(T) and define C_{∞}(T) as the limit of these extensions. In this way we can get a extended central closure which is really a closure. Nevertheless, we will only make use of the first step C(T)T since it will be enough for our purposes.

5. Jordan systems with nonzero PI-elements

In this section we prove our main result, wich describes strongly prime Jordan systems having nonzero PI-elements through its extended central closure:

5.1 Theorem. Let J be a strongly prime Jordan system. If PI(J) ≠ 0, then the extended central closure C(J)J of J has nonzero socle equal to PI(C(J)J), and PI(J) = J ∩ Soc(C(J)J).

As commented before, this theorem should be considered as a Jordan analogue of Martindale’s Theorem on prime associative algebras satisfying a generalized identity. Since its proof is somewhat involved, we first sketch the argument, and we will devote the rest of this section to the proof.

5.2 We first change our notation and write J_{0} for the original system and J for its extended central closure. The theorem asserts that J is rationally primitive, and according to 0.17 this is equivalent to the existence of an element a ∈ J whose local algebra J_{a} is simple and PI. To prove that there is such an element in J, suppose first that there is a ∈ PI(J) such that J_{a} is strongly prime. By the Jordan analogue 0.13 of Posner’s theorem the central closure of J_{a} is simple and unital, so it suffices to show that the centroid Γ_{a} = Γ(J_{a}) is already a field. Moreover, since J_{a} is an algebra over the extended centroid C(J_{0}) of J_{0}, and Γ_{a} is a domain (J_{a} being strongly prime), it suffices to show that Γ_{a} is algebraic over C(J_{0}).

To prove that claim we use Amitsur’s ultraproduct construction much in the same spirit as was used by Martindale [19]. Thus we find a primitive embedding J ⊆ J̃ with PI(J) = PI(J̃) ∩ J. The system J̃ is then rationally primitive, hence if J is an pair or an algebra, the local J̃_{a} is a simple and unital PI-algebra (we will prove that below). In particular, its centroid Γ̃_{a} is a field.

Now, if J̃ is a system over a large field Ω, the extension Γ̃_{a} will be algebraic by Amitsur’s trick, so if Ω is algebraically closed, we will have Γ̃_{a} = Ω = Γ(J̃). Therefore we can look for inverses of the elements of Γ_{a} in Γ(J̃). Here is where the key property 4.4 of the extended centroid aplies. We can capture a power of the inverses back into C(J), and by 4.6, back into C(J_{0}), thus yielding inverses for some elements of Γ_{a}, and algebraicity of Γ_{a} over C(J_{0}).

Two remarks are in order in view of this plan. First, at same point in this argument, we had to assume that J is an algebra or a pair. The problem for triple systems is that the local J̃_{a} need not be primitive, and this will require a separate argument using tight doubles.

Second, we will not really find a primitive J̃ over a large algebraically closed field Ω > dim_{Ω}J̃ + 2, since the ultrafilter construction makes it difficult to control dim J̃. Instead we will have largeness of Ω with respect to the factors J_{λ} of J̃, but this will suffice since we will be able to lift inverses in each factor J_{λ} to a global inverse in J.

5.3 We start with the construction of a primitive embedding for an strongly prime system J with nonzero PI-elements. Following Amitsur’s construction, we first need to get rid of the Jacobson radical.

Let J be a Jordan system (triple, algebra or pair). We denote by PNil(J) the properly nil radical of J. The set PI(J) of elements a ∈ J such that the local algebra J_{a} is PI. Recall [24] that if J is nondegenerate this is an ideal of J. The following result is Theorem 6.3 of [24].

5.4 Theorem. Let J be a nondegenerate Jordan system, then PI(J)∩PNil(J) = 0. In particular, if J is prime then PI(J) ≠ 0 implies PNil(J) = 0.

Now we cat get a semiprimitive system from a strongly prime J by means of a scalar extension.

5.5 Lemma. Let J be a strongly prime Jordan system over a large field Ω: | Ω |> dim_{Ω}J + 2. If PI(J) ≠ 0, then J is semiprimitive.

Proof: Since PI(J) ≠ 0 implies PNil(J) = 0 by 5.4, this follows from 8.6 of [2].

Using this result, we can get a scalar extension of J which is a subdirect sum of primitive systems. The embedding we are looking for will result as an ultraproduct of that familly of primitive images. To deal with it we first study ultraproducts of the kind of Jordan systems we are interested in.

5.6 Lemma. Let (J_{λ})_{λ∈Λ} be a familly of Jordan systems, let F be an ultrafilter on Λ, and let J̃ = ∏ J_{λ}/F be the corresponding ultraproduct. If the set {λ ∈ Λ | J_{λ} is strongly prime with nonzero socle } belongs to F, then J̃ is strongly prime with nonzero socle.

Proof: Put Λ_{0} = {λ ∈ Λ | J_{λ} is strongly prime with nonzero socle }

Nondegeneracy is expressed by the elementary sentence

∀x ∃y, x ≠ 0 ⇒ P_{x}y ≠ 0,

and since it holds in J_{λ} for all λ ∈ Λ_{0}, and Λ_{0} ∈ F, it holds in J̃ (see [26]). Therefore J̃ is nondegenerate.

Next, each J_{λ} with λ ∈ Λ_{0} has nonzero socle, hence it contains a simple element by nondegeneracy (see [17]). For nondegenerate systems this means that the elementary sentence

∃x ∀y ∃z, x ≠ 0 & (P_{x}y ≠ 0 ⇒ P_{P_{x}y}z = x),

holds in each J_{λ} for λ ∈ Λ_{0}. Therefore it holds in J̃, and J̃ has nonzero simple elements, hence it has nonzero socle.

Finally, primeness follows from the elementary characterization 0.3, since the each J_{λ} with λ ∈ Λ_{0} is elementary prime, hence so is J̃, and thus it is strongly prime.

Note that the full strength of the results of [5] implies that the ultraproduct of strongly prime algebras is again strongly prime.

5.7 Let us return to the situation we are interested in. After our change of notation, J_{0} is the original system and J = C(J_{0})J_{0} is its central closure. Take Ω an algebraically closed field containing C(J_{0}) with cardinality | Ω |> dim_{C(J_{0})}J + 2, and form a tight scalar extension ΩJ. By 5.5, Jac(ΩJ) = 0 (since dim_{Ω}ΩJ ≤ dim_{C(J_{0})}J), hence by 0.8, ΩJ is a subdirect product of its primitive images (J_{λ})_{λ∈Λ}. Let π_{λ} : ΩJ → J_{λ} be the corresponding projection, and consider the directed system of sets Λ_{x} = {λ ∈ Λ | π_{λ}(x) ≠ 0}, for 0 ≠ x ∈ ΩJ . Let F be an ultrafilter containing the system {Λ_{x} | 0 ≠ x ∈ ΩJ}, and form the ultraproduct J̃ = ∏ J_{λ}/F, which is again a Jordan system of the same type (pair, triple or algebra) over the field Ω̃ = Ω^{Λ}/F, the ultrapower of Ω. Notice that J embeds in J̃ (see [31, 2]), so we will consider J as a subsystem of J̃.

We next show that PI-elements of J are still PI-elements of J̃. This follows from the following extension principle 5.8 Lemma. Any generalized polynomial

p(x_{1}, …, x_{n}) = f(x_{1}, …, x_{n}, b_{1}, …, b_{m})

for fixed b_{i} ∈ J and f(x_{1}, …, x_{n}, y_{1}, …, y_{m}) of the free Jordan system F T [X ∪ Y] has: f(ΩJ, …, ΩJ, b_{1}, …, b_{m}) = 0 implies f(J̃, b_{1}, …, b_{m}) = 0.

Proof: This follows since the vanishing of f(x_{1}, …, x_{n}, b_{1}, …, b_{m}) for all x_{i} is an elementary sentence that is inherited by quotients of ΩJ.

Notice also that the condition on f holds if for any linearization g of f in x_{1}, …, x_{n} we have g(J, …, J, b_{1}, …, b_{m}) = 0. In particular, if f is linear in the variables x_{i}, then f(J, …, J, b_{1}, …, b_{m}) = 0 suffices.

5.9 Let a be a nonzero element in PI(J), then J_{a} satisfies some multilinear identity f(x_{1}, …, x_{n}) = 0, hence it satisfies f(a; J, …, J) = 0 where f(z; x_{1}, …, x_{n}) is the z-homotope of f. By the previous lemma, J̃ still satisfies f(a; J, …, J) = 0, and J̃_{a} has the PI f. Thus we get PI(J) ⊆ J ∩ PI(J̃). Since the reverse containment is obvious, we obtain PI(J) = J ∩ PI(J̃).

Also, for each λ ∈ Λ_{a},the projection of a into J_{λ} is a nonzero PI-element, and since J_{λ} is primitive, it is strongly prime with nonzero socle by 0.17. Thus Lemma 5.6 implies that J̃ is strongly prime with nonzero socle equal to PI(J̃). In particular, J̃ is primitive at each a ∈ PI(J) by 0.11.

Now, according to the plan sketched in 5.2, he have to choose a nonzero a ∈ PI(J) such that J_{a} is prime. This follows as an straightforward application of the results of [5], however, since these results make use of what was proved in [2], and we are interested in making our results independent of those, we give an alternative, rather ad hoc approach. For it we use the following

5.10 Lemma. If b̄_{1}, b̄_{2} ∈ J_{a} generate orthogonal ideals: id_{J_{a}}(b̄_{1}) ∩ id_{J_{a}}(b̄_{2}) = 0, then they generate orthogonal ideals in J̃_{a}. In particular, if J̃_{a} is prime, then J_{a} is prime.

Proof: We prove this for Jordan triple systems, although the same proof works for pairs and algebras.

Suppose that c_{i} ∈ J̃ has c̄_{i} = c_{i} + Kera ∈ idJ̃_{a}(b̄_{i}), for i = 1, 2, then there are algebra polynomials q_{i} = q_{i}(x_{i1}, …, x_{in_{i}}, y_{i}) for i = 1, 2, all of whose monomials contain the variable y_{i}, such that c̄_{i} = q_{i}(a; d_{i1}, …, d_{in}, b_{i}) + Kera for some d_{ij} ∈ J̃. Then, setting

g(x_{11}, …, x_{1n_{1}}, x_{21}, …, x_{2n_{2}}, y_{1}, y_{2}, y_{3}) = P_{y_{3}}P_{q_{1}}P_{y_{3}}q_{2},

we get g(ΩJ, …, ΩJ, b_{1}, b_{2}, a) = 0 since q_{i}(ΩJ, …, Ω, b_{i}, a) + Kera ∈ id_{ΩJ_{a}}(b̄_{i}), and id_{ΩJ_{a}}(b̄_{1}) ∩ id_{ΩJ_{a}}(b̄_{2}) = 0. Thus, by 5.8, g(T̃, b_{1}, b_{2}, a) = 0, hence P_{a}P_{c_{1}}P_{a}c_{2} = 0, i.e. Uc̄_{1} c̄_{2} = 0.

Therefore U_{idJ̃a (b̄_{1})}idJ̃_{a}(b̄_{2}) = 0, and idJ̃_{a}(b̄_{1}) ∩ idJ̃_{a}(b̄_{2}) = 0 by nondegeneracy of J_{a}.

5.11 Returning to our search of a, by the fact just proved, for J_{a} to be prime it suffices that J̃_{a} be prime, but if our systems are algebras or pairs, J̃_{a} will be primitive for any a ∈ Soc(J̃)∩J by 0.9 and 0.11, hence for any a ∈ PI(J̃)∩J = PI(J). Moreover, if λ ∈ Λ_{a}, and a_{λ} is the image of a in J_{λ}, then (J_{λ})_{a_{λ}} is a simple algebra (by the same reasons as J̃_{a}.)

For triple systems the situation is not as neat as for pairs or algebras due to the fact that local algebras of primitive triple systems need not be primitive (see[1]). This forces us to split the proof and consider separately Jordan triple systems. Before treating them, we first focus on pairs and algebras (here P denotes the operators Q and U of pairs and algebras respectively, and we reserve the U-notation for local algebras).

5.12 End of the proof of 5.1 for pairs and algebras.

According to 5.11, we can take a nonzero a ∈ PI(J), such that J_{a} is strongly prime. Since it is PI, by 0.13 its central closure Γ_{a}^{−1}J_{a}, where Γ_{a} = Γ(J_{a}) is the centroid of J_{a}, is simple and unital, equal to its socle. Our aim is to show that Γ_{a} is in fact a field (so that J_{a} is its own central closure, and therefore it is simple, equal to its socle). To see that we show that Γ_{a} is algebraic over C(J_{0}).

We can find γ ∈ Γ_{a} and z̄ ∈ J_{a} such that 1 = γ^{−1}z̄ is the unit element of J_{a}. In particular, Uz̄ = U_{γ(1)} belongs to Γ_{a}. This shows that there are nonzero elements z̄ ∈ J_{a} such that Uz̄ belongs to the centroid Γ_{a}. We take z̄ any of those elements. Then, for all λ ∈ Λ_{a}, the operator Uz̄_{λ}, where z_{λ} is the image of z in J_{λ}, belongs to the centroid Γ_{a_{λ}} of (J_{λ})_{a_{λ}}. Since this is a simple algebra over Ω, Γ_{a_{λ}} is a Ω-field. Also, | Ω |> dim_{Ω}ΩJ + 2 ≥ dim_{Ω}J_{λ} + 2 ≥ dim_{Ω}(J_{λ})_{a_{λ}} + 2. Thus, by Amitsur’s trick (see [2]), Γ_{a_{λ}} = Ω, since Ω is algebraically closed. Thus there is μ_{λ} ∈ Γ_{a_{λ}} such that Uz̄_{λ} x̄ = μ_{λ}x̄ for all x̄ ∈ (J_{λ})_{a_{λ}}. Therefore P_{P_{a_{λ}} z_{λ}}x_{λ} = μ_{λ}P_{a_{λ}}x_{λ} for all x_{λ} ∈ J_{λ}.

Now, the choice of the previous μ_{λ} for each λ ∈ Λ_{a}, and μ_{λ} = 0 otherwise, defines an element μ ∈ Ω̃ which satisfies P_{P_{a}z}x = μP_{a}x for all x ∈ J̃ (since this is an elementary sentence holding in each J_{λ}). In particular, if we take x ∈ J we get μP_{a}x = P_{a}P_{z}P_{a}x ∈ J, and μP_{a}J ⊆ J. Thus, by Lemma 4.5, there is a permissable map (f, I) of J such that for all y ∈ I, f(y) = μy. Now, by 4.6, C(J) = C(C(J_{0})J_{0}) is extension of C(J_{0}) of exponent two, hence there is δ ∈ C(J_{0}) such that for all y ∈ I ⋆J, f^{2}(y) = δy. Then, μ^{2}x = δx for all x ∈ J. In particular in the algebra J_{a} we have, (U 2 ; z̄ − δ)x̄ = 0, and Uz̄ is algebraic over C(J_{0}) for all z̄ such that Uz̄ ∈ Γ_{a}. Now, if τ ∈ Γ_{a} and 0 ≠ Uz̄ ∈ Γ_{a}, then there are ρ, δ ∈ C(J_{0}) such that U_{τz̄}^{2} = ρ and U 2 ; z̄ = δ. Thus τ^{2}δ = τ^{2}U 2 ; z̄ = U_{τz̄} = ρ, and τ is algebraic over C(J_{0}).

Therefore, Γ_{a} is algebraic over C(J_{0}), and since the latter is a field and Γ_{a} is a domain (since J_{a} is prime), this must be a field too.

5.13 Remark. What we have proved is that if J = J_{0}(J_{0}) is a Jordan pair or Jordan algebra, then for any a ∈ PI(J), such that J_{a} is strongly prime, the local centroid Γ_{a} = Γ(J_{a}) is algebraic over C(J). As said before, this implies that J_{a} is a simple PI-algebra, hence J is strongly prime with PI(J) = Soc(J). Then, for any nonzero b ∈ PI(J) = Soc(J), J is primitive at b (see 4.3 of [24]), and J_{b} is prime (and, in fact primitive by 6.1 of [1]). Thus, the previous argument can be applied to b instead of a, and we conclude that all local centroids Γ(J_{b}) at elements b ∈ PI(J) are algebraic over C(J).

We now turn to Jordan triple systems, for which we first need some perparatory work.

5.14 We say that a Jordan triple system J is of polarized type if there are nonzero semiideals I, L of T such that I ∩ L = 0. Equivalently, J is of polarized type if its double V (J) is not tight (see Lemma 2.4 of [5]).

5.15 Lemma. Let J be a primitive Jordan triple system with nonzero socle, and let 0 ≠ a ∈ Soc(J). Then J_{a} is simple with capacity if J is not of polarized type, and J_{a} is a direct sum of at most two simple algebras with capacity if J is polarized. In particular J_{a} is prime if and only if it is simple.

Proof: Suppose first that J is not of polarized type, then V (J) is a tight double of J. Moreover, V (J) is primitive at any element of Soc(V (J)) = V (Soc(J)) (see [24]). Now, a ∈ Soc(J) can be considered an element of Soc(V (J))^{−}. Therefore J_{a} = V (J)_{a}^{+} is primitive by Theorem 6.1 of [1], and it has capacity by 0.6 of [24].

Next suppose that J is of polarized type, and take W = V (J)/I a tight double of J. Set a^{σ} = a + I^{σ} ∈ J/I^{σ} = W^{σ}. Then a^{σ} ∈ Soc(W^{σ}) by 4.2 of [24], and W is primitive both at a^{+} and a^{−} by 0.11. Thus the algebras W_{a^{−σ}}^{σ}, for σ = ±, are primitive with capacity by 6.1 of [1] and 0.6 of [24]. Since J_{a} is a subdirect sum of W_{a^{−}}^{+} and W_{a^{+}}^{−} by [1], it is a full direct sum of these algebras.

5.16 Lemma. Let J be a primitive Jordan triple system with nonzero socle, and let a ∈ Soc(J). If Ī_{1} and Ī_{2} are nonzero ideals of J_{a} which are orthogonal: Ī_{1}∩Ī_{2} = 0, then for any nonzero b̄ = b + Kera ∈ Ī_{1}, the local algebra J_{P_{a}b} is primitive.

Proof: Since, by hypothesis, J_{a} is not prime, 5.15 implies that J_{a} is a subdirect sum of two primitive algebras J_{1}, J_{2}. Now, if π_{i} : T_{a} → J_{i} is the corresponding projection, we have U_{π_{i}(Ī_{1})}π_{i}(Ī_{2}) = 0, hence either π_{i}(Ī_{1}) = 0 or π_{i}(Ī_{2}) = 0 by strong primeness of J_{i}. Thus we can assume that π_{1}(Ī_{1}) = 0. Then, (J_{a})b̄ = (J_{a}/Kerπ_{1})_{b̄+Kerπ_{1}} = (J_{2})b̄ since Ī_{1} ⊆ Kerπ_{2} and Kerπ_{1} ∩ Kerπ_{2} = 0. But (J_{2})b̄ is primitive by [8].

Now we return to the notation introduced before, so that J is a Jordan triple system and J̃ is a ultraproduct of the primitive images J_{λ}, λ ∈ Λ of the scalar extension ΩJ.

5.17 Lemma. With the previous notation, if J̃_{a} is primitive for a ∈ PI(J), then the set Λ_{0} of λ ∈ Λ such that (J_{λ})_{a_{λ}} is primitive belongs to the ultrafilter F.

Proof: In this case J̃_{a} is simple with capacity, and by 5.15, all algebras (J_{λ})_{a_{λ}} have capacity, and are either simple or a direct sum of two simple algebras. Both conditions can be expressed by elementary sentences since (J_{λ})_{a_{λ}} is simple if and only if its double V ((J_{λ})_{a_{λ}}) is simple, and here simplicity is equivalent to the fact that any two minimal idempotents are conected (see [17]). It is straightforward that all this can be expressed by means of elementary sentences. Thus, if Λ_{1} is the set of indexes λ ∈ Λ_{a} such that (J_{λ})_{a_{λ}} is not simple, one of the sets Λ_{0}, Λ_{1} belongs to F. If Λ_{1} ∈ F, then J̃_{a} can not be simple since the corresponding elementary sentence holds in J̃. Therefore Λ_{0} belongs to F.

5.18 Lemma. With the previous notation, if either J is of polarized type or J̃ is not of polarized type, then there is a nonzero element a ∈ PI(J) such that J̃_{a} and J_{a} are prime. In any case there is a ∈ PI(J) such that J_{a} is prime.

Proof: Suppose first that J̃ is not of polarized type. Then PI(J̃) = Soc(J̃) by 0.17, J̃_{a} is prime by 5.15, and J_{a} is also prime by 5.10.

If J is of polarized type, take (I^{+}, I^{−}) a nonzero ideal of V (J) with I^{+} ∩I^{−} = 0. Note that PI(J) is an ideal by 0.16, and for σ = ±, P_{I^{σ}}PI(J) ⊆ I^{σ} ∩PI(J) is nonzero, since otherwise I^{σ} would annihilate PI(J) by 1.7 of [20], contradicting primeness of J. Thus we can take nonzero elements a^{σ} ∈ PI(J) ∩ I^{σ} and set a = a^{+} + a^{−}. Then J_{a} has nonzero ideals Ī^{σ} = I^{σ} + Kera/Kera for σ = ±, and it is easy to see that they are orthogonal Ī^{+} ∩ Ī^{−} = 0. Take nonzero elements b̄^{σ} = b^{σ} + Kera. Then they generate orthogonal ideals in J̃ by 5.10, hence the algebras J̃_{P_{a}b_{σ}} are prime by 5.16, and so are the algebras J_{P_{a}b^{σ}} by 5.10. Also P_{a}b_{σ} ∈ PI(J) since PI(J) is an ideal and a ∈ PI(J).

Finally, notice that if 0 ≠ a ∈ PI(J), and J_{a} is not prime, we can apply the same argument as before to get an element P_{a}b such that J_{P_{a}b} is prime, so we can always find a nonzero element a ∈ PI(J) such that J_{a} is prime.

5.19 The previous result will allow us to apply the same argument as in 5.12 to the proof of Theorem 5.1 in the cases where J is of polarized type or neither J nor J̃ are of polarized type. However it might happen that J is not of polarized type, but J̃ is polarized. We will deal with this case by applying the already proved results to the double V (J). To do that we need some information on the extended centroid of V (J) (which is a field since V (J) is strongly prime if J is not of polarized type).

For a Jordan triple system J we can define a mapping C(J) → C(V (J)) by simply doubling J-homomorphisms: If λ = [f, I], then ((f^{+}, f^{−}), (I^{+}, I^{−})), where f^{+} = f^{−} = f and I^{+} = I^{−} = I, defines a permissable mapping of V (J), and therefore, an element of C(J). It is easy to see that this is a well-defined monomorphism of rings, so that we can view C(J) as contained in C(V (T)).

5.20 Lemma. Let J be a strongly prime Jordan system, and suppose that it is not of polarized type, so that V (J) is a tight double. Then C(J) ⊆ C(V (J)) is a degree two extension of fields.

Proof: If λ = [(f^{+}, f^{−}), (I^{+}, I^{−})] is an element of C(V (J)), we define λ^{∗} as the class of ((f^{−}, f^{+}), (I^{−}, I^{+})). This is easily seen to be a well-defined mapping, and it is straightforward to check that this is an automorphism of C(V (J)) of period two.

Now, if λ = λ^{∗} is a ∗-invariant element of C(J), and ((f^{+}, f^{−}), (I^{+}, I^{−})) is a representative of λ, then ((f^{+}, f^{−}), (I^{+}, I^{−})) and ((f^{−}, f^{+}), (I^{−}, I^{+})) are equivalent permissabe mappings, hence (g, L), where L = I^{+} ∩ I^{−} and g = f_{L}^{+} = f_{L}^{−} (see 1.8), is a permissable map of J, and it is clear that λ = [g, L] under the identification C(J) ⊆ C(V (J). Thus, the set of ∗-invariant elements of C(V (J)) is precisely C(J), and this proves the lemma.

5.21 End of the proof of 5.1 for triple systems.

By 5.18, if J is of polarized type (or if J̃ is not of polarized type), we can find a nonzero element a ∈ PI(J) such that both J̃_{a} and J_{a} are prime. Then, by 5.17, the set Λ_{0} of indexes λ ∈ Λ such that (J_{λ})_{a_{λ}} is prime (hence simple and PI by 5.15) belongs to F. Then the proof given in 5.12 for pairs and algebras applies verbatim if we take Λ_{0} instead of Λ_{a}.

Next, if J is not of polarized type, the double V = V (J) is a strongly prime Jordan pair. Now, by the last assertion of 5.18, there is a nonzero a ∈ PI(J) such that J_{a} is prime (hence strongly prime, since nondegeneracy is obviously inherited by local algebras). Then, setting a^{σ} = a ∈ V^{σ}, the local V_{a^{−}}^{+} = J_{a} is PI, and therefore V has PI-elements and we can apply what was proved in 5.12 (see also 5.13) to conclude that the local centroids Γ(C(V)V_{b}) of C(V) at PI-elements b ∈ PI(C(V)V) are algebraic over C(V). Now, since C(V) is algebraic over C(J) by 5.20, and C(J) is algebraic over C(J_{0}) by 4.6, the field Γ(C(V)V_{b}) is algebraic over C(J_{0}).

Finally note that C(V)V_{a^{−}}^{+} is a central extension of V_{a^{−}}^{+} = J_{a}, and therefore Γ_{a} = Γ(J_{a}) embeds canonically into Γ(C(J)J_{a}). Therefore Γ_{a} is algebraic over C(J_{0}), and since it is a domain, it must be a field. Thus again we conclude that J_{a} is its own central closure, and since it is PI, it is simple.

6. Homotope polynomial Identities

As we have already mentioned, D’Amour and McCrimmon’s work [1, 2] suggest that an operant PI-theory for general Jordan systems should deal with homotope identities rather than identities (polynomials of the free system which vanish under substitutions but are nonzero in the free special system). In [24] we proved 0.18, an analogue of Kaplansky’s theorem for homotope-PIs on primitive Jordan systems. As a consequence of our main theorem we can prove here an analogue of Posner’s theorem for general Jordan systems:

6.1 Theorem. Let J be a prime nondegenerated Jordan system.

(i) If the local algebra at each element of J is PI, then the extended central closure C(J)J is simple equal to its socle.

(ii) If J satisfies a homotope-PI, then C(J)J is simple with capacity.

Proof: In both cases C(J)J is primitive by 5.1 and we can apply 0.18.

Admitedly, this is a weak version of Posner’s theorem, or rather of the sharper Posner-Rowen’s theorem, since here we need the extended central closure instead of the (ordinary) central closure Γ^{−1}J. If J is a Jordan algebra, we now 0.13 that the stronger version holds, but there is a basic reason for finding an obstrucion in its extension to Jordan systems, namely, there cannot be central polynomials in Jordan systems other than algebras. A possible substitute would be central polynomials of the multiplication algebra, and indeed this detour works for exceptional systems (see [30, p. 517] and [2]). However this relies on the outer-simplicity of exceptional systems, hence there is not a straightforward way of proving their general existence. Nevertheless, we state it here as a conjecture:

6.2 Conjecture 1. The extended centroid of a strongly prime homotope-PI Jordan system coincides with the field of fractions of its centroid.

6.3 In spite of the preference for homotope-polynomial identities mentioned before, Theorem 2 of [30] suggest that a general PI-theory of Jordan systems could be worked out. On the other hand, it was proved in [24] that any PI Jordan algebra satisfies a homotope-PI. Call a Jordan system J a PI-system if there is an essential polynomial which vanishes under all substitutions of its variables, where a polynomial is essential if it is not a special identity. We state

6.4 Conjecture 2. Every PI-Jordan system satisfies a homotope PI.

6.5 Theorem 5.1 will be used in [13] to relate inner ideals of J a special Jordan algebra and one-sided ideals of its ∗-tight envelopes. Here, however, we give an application which illustrates the srtength of that result. We generalize D’Amour and McCrimmon’s Primitive Form Theorem (Theorem 5.6 of [DAMcl], which states that if a simple Jordan pair V is a scalar form of some classical pair Ṽ of Clifford type, then V is primitive.

6.6 Theorem. If a simple Jordan system J with PI(J) ≠ 0 is a scalar form of some nondegenerate J̃, then J is primitive. Moreover, if J satisfies a homotope-PI, then it has capacity.

Proof: We only need J̃ to pass nondegeneracy down to J, so that J is strongly prime, and C(J)J is primitive by 5.1. Now C(J) = Γ(J) since J is simple, so J = Γ(J)J is primitive. If, in addition J satisfies a homotope-PI, we can apply 0.18 to conclude that J has capacity.

(A different approach would be as follows: If Jac(J) = J, a suitable tight scalar extension J′ is properly nil (see [2, 8.6]), and then it is McCrimmon radical by [24, 6.3(b)], which is impossible since J ⊆ J′ is tight, and J is nondegenerate.)

6.7 As a final remark let us point out that 6.1 improves Zelmanov-D’Amour-McCrimmon’s classification of strongly prime Jordan systems (see [2] or [6]): the extended central closure of Clifford, bi-Cayley and Albert forms is already Clifford, bi-Cayley or Albert (instead of the less definite scalar extensions required in the original proofs).

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