Outer Inheritance of Simplicity in Jordan Systems

José A. Anquela ^{1}, Teresa Cortés ^{1}, anque@pinon.ccu.uniovi.es, cortes@pinon.ccu.uniovi.es Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain

Esther García ^{1,} ^{2}, egg@pinon.ccu.uniovi.es Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Avenida Complutense s/n, 28040 Madrid, Spain

and

Kevin McCrimmon kmm4m@virginia.edu Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.

Abstract: In this paper we show that outer ideals of simple Jordan systems have simple cube. In most cases we obtain that the outer ideals inherit simplicity and give precise descriptions in the cases in which simplicity is not obtained.

Introduction

Outer ideals of Jordan systems appear naturally in the classifications of strongly prime, primitive, simple and division Jordan systems when characteristic two is allowed [1, 3, 4, 22]: Ample hermitian subspaces H_{0}(A, ∗) of an associative system A with involution (see [22, 0.8’]) are outer ideals of the Jordan system of symmetric elements H(A, ∗), while some of the classes of simple and division Jordan algebras are described as “ample” outer ideals of other algebras [22, 15.5, 15.7].

In [21] the transfer of regularity conditions between a Jordan algebra and its ample outer ideals is studied. In [5], some of the results of [21] are extended to nonnecessarily ample outer ideals of arbitrary Jordan systems. However, no new results on the transfer of simplicity are obtained in [5], so that only the inheritance of simplicity by ample outer ideals of unital Jordan algebras [21, 5.1] is known.

1 Partially supported by the Ministerio de Ciencia y Tecnología, BFM2001-1938-C02-02.

2 Partially supported by an F.P.I. Grant (Ministerio de Ciencia y Tecnología).

We show that if K is a nonzero outer ideal of a simple Jordan pair or triple system then K^{3} is simple. The analogous result is obtained for simple Jordan algebras and squared outer ideals. Moreover, we describe precisely the case when K itself is not already simple. Using hermitian polynomials (cf. [7, Sect. 1]) and [21, 5.1], our problem is reduced to that of the simplicity of an ample hermitian subspace of a simple associative system, which explains the form of our result, patterned after [7, 2.7(ii), 3.7(ii), 3.15(ii)], hence directly linked to the open problem of the inheritance of simplicity by ample hermitian subspaces.

The paper is divided into five sections, the first of which is devoted to displaying basic terminology and known results, as well as to proving some elementary auxiliary facts. After that, we study simple PI Jordan systems: we show that they are von Neumann regular, which will be used, by means of local algebras, to reduce the problem of pairs and triple systems to unital algebras. In Section 2 we obtain our main theorem on algebras, which is based on the classification theorem of simple Jordan algebras [22, 15.5]. We begin by showing that outer ideals of simple unital algebras are automatically ample as soon as they are subalgebras (i.e., closed by squares); this improves [21, 5.1] and reduces our problem to the non-PI case. We manage to solve this case in the three kinds of Jordan systems by using hermitian polynomials. In Section 3, we study the simplicity of outer ideals in the presence of an HPI, which completes the solution to our problem in the case of pairs and triple systems. We use local algebras and the results of the previous section in the pair case, and obtain the triple system result through the functor V. The final section deals with Jordan pairs covered by a connected ortho-collinear grid. Their specific forms allow a direct study of their outer ideals.

0. Preliminaries

0.1 We will deal with associative and Jordan algebras, pairs, and triple systems over an arbitrary ring of scalars Φ. We will follow closely the terminology of [5], and refer also to [4, 10, 16, 22].

Recall that the products x^{2}, U_{x}y of x, y in a Jordan algebra J are quadratic in x and linear in y, and have linearizations x ∘ y = V_{x}y, U_{x,z}y = V_{x,y}z = {x, y, z}, respectively. The unital hull J + Φ1 of J will be denoted by Ĵ. The products in a Jordan triple system have the form P_{x}y, with linearizations P_{x,z}y = L_{x,y}z = {x, y, z}. In a Jordan pair V = (V^{+}, V^{−}) the products have the form Q_{x}y, with linearizations Q_{x,z}y = D_{x,y}z = {x, y, z}, for any x, z ∈ V^{σ}, y ∈ V^{−σ}, σ = ±.

0.2 The following identities hold in any Jordan algebra and will be needed in the sequel:

(i) {x, y, U_{a}b} = {{x, y, a}, b, a} − U_{a}{y, x, b}.
(ii) x ∘ U_{a}b = {x ∘ a, b, a} − U_{a}(x ∘ b).
(iii) (a ∘ b) ∘ c = {a, b, c} + {b, a, c}.
(iv) U_{U_{a}b}z = U_{a}U_{b}U_{a}z, (U_{a}b)^{2} = U_{a}U_{b}a^{2}.
(v) x ∘ U_{a}b = {x, b ∘ a, a} − {x, b, a^{2}}.
(vi) U_{x}U_{a}b = U_{x∘a}b + U_{U_{x}a,a}b − U_{a}U_{x}b − x ∘ U_{a}(x ∘ b).
(vii) (a ∘ b)^{2} = U_{a}b^{2} + U_{b}a^{2} + b ∘ U_{a}b.

[The first one is JP12 of [15], and the rest of them hold in all special Jordan algebras and are either two variable identities or three variable identities linear in at least one of them, hence they also hold in any Jordan algebra by Macdonald’s Theorem [15]. Note (ii) results by setting y = 1 in (i); (iv) is the Fundamental Formula and its application to z = 1; and (vi) follows by replacing a variable by 1 in one of the linearizations of U_{x}U_{a}U_{x}b = U_{U_{x}a}b.]

0.3 A Jordan algebra J gives rise to a Jordan triple system by simply forgetting the squaring and letting P = U; a Jordan triple system T gives rise to a double Jordan pair V (T) = (T, T) by setting Q = P; a Jordan pair V = (V^{+}, V^{−}) yields a polarized Jordan triple system T (V) = V^{+} ⊕ V^{−} by defining P_{x^{+}⊕x^{−}}(y^{+} ⊕ y^{−}) = Q_{x^{+}}y^{−} ⊕ Q_{x^{−}}y^{+} [16, 1.13, 1.14].

Similarly, one can consider the underlying triple system of an associative algebra, as well as functors V ( ) and T ( ) between the categories of associative pairs and associative triple systems.

0.4 Recall how the functors V ( ) and T ( ) preserve simplicity:

(i) A Jordan or associative pair V is simple if and only if T (V) is simple. An associative pair V with involution ∗ is ∗-simple if and only if T (V) is ∗-simple (cf. [1, p. 230] and [7, 0.5]).

(ii) A Jordan triple system T is simple if V (T) is simple (cf. [6, 0.3]).

0.5 We recall the notion of inner ideal of a Jordan system. A Φ-submodule K of a Jordan algebra J is called an inner ideal of J if U_{K} Ĵ ⊆ K, i.e., U_{K}J + K^{2} ⊆ K. An inner ideal K of a Jordan triple system T is a Φ-submodule of T satisfying P_{K}T ⊆ K, and an inner ideal K of a Jordan pair V is a Φ-submodule of V^{σ} (σ = + or −) satisfying Q_{K}V^{−σ} ⊆ K. Inner ideals of Jordan systems are always subsystems (subalgebras, subtriples, subpairs).

0.6 Let us recall the notion of outer ideal K of a Jordan system J (denoted K ◁_{out} J). A Φ-submodule K of a Jordan algebra J is called an outer ideal of J if it remains invariant under outer multiplications in the unital hull: UĴK ⊆ K, i.e., U_{J}K + V_{J}K ⊆ K (which implies V_{Ĵ,Ĵ}K ⊆ K by (0.2)(iii)). An outer ideal K of a Jordan triple system T is a Φ-submodule of T satisfying P_{T} K + L_{T,T} K ⊆ K, and an outer ideal of a Jordan pair V is a pair of Φ-submodules K = (K^{+}, K^{−}) satisfying Q_{V^{σ}}K^{−σ} +D_{V^{σ},V^{−σ}}K^{σ} ⊆ K^{σ} (σ = ±). An ideal of a Jordan algebra or triple system is both an inner and an outer ideal. An outer ideal K of a Jordan pair V is an ideal of V if both K^{+} and K^{−} are inner ideals of V. Notice that if 1/2 ∈ Φ, then outer ideals are just ideals.

In Jordan algebras the unital hull (with the unit element and its accomplice, the square) is always lurking behind the scenes, and it is natural to consider imposing two extra conditions on outer ideals.

For triple systems and pairs, outer ideals are always subsystems, but for algebras outer ideals need not be subalgebras; they are just triple subsystems of the underlying triple system [21, 0.3] (we have 2K^{2} = K ∘ K ⊆ K, but not necessarily K^{2} ⊆ K). Since we are primarily interested in properties inherited by K, which usually make sense only if K itself is an algebra, we will focus most of our attention on squared outer ideals: those outer ideals K which are also subalgebras or, equivalently, which are closed under squares

U_{J}K + V_{J}K + K^{2} ⊆ K.

We will almost always impose this first condition on algebra outer ideals.

The second condition we sometimes impose is that the outer ideal contains the unit (or at least its ghost). An outer ideal K of a Jordan algebra J is called ample if J^{2} + 2J ⊆ K (cf. [21, 1.2]); all ample outer ideals are squared. Ample outer ideals are big chunks of J: if 1/2 ∈ Φ then the only ample outer ideal is J itself. An outer ideal K in a unital Jordan algebra J is ample if and only if it contains 1 (in which case we call it unitally ample). Indeed, a Φ-submodule K of J is a unitally ample outer ideal if and only if 1 ∈ K and U_{J}K ⊆ K (cf. [21, 1.1]). An outer ideal K is ample in a non-unital J if and only if it is the “shadow” K = J ∩ K̂ of a “ghost” ample outer ideal K̂ of the unital hull Ĵ (cf. [21, 1.2]). Notice that in triple systems and pairs there is no concept of ampleness for outer ideals.

0.7 Ample outer ideals are closely related to ample hermitian subspaces. For any associative triple system R with involution ∗, an ample hermitian subspace of (R, ∗) is a Φ-submodule H_{0} = H_{0}(R, ∗) of H(R, ∗) closed under traces and “starred P-operators”:

x + x^{∗} ∈ H_{0}, xH_{0}x^{∗} ⊆ H_{0} for all x ∈ R.

For an associative pair R = (R^{+}, R^{−}), we will only consider polarized involutions ∗, i.e., involutions of the triple system T (R) which respect the polarization [(R^{σ})^{∗} ⊆ R^{σ}] (see [9, 1.7]). Given an associative pair R = (R^{+}, R^{−}) with polarized involution ∗, an ample hermitian subspace is a pair of Φ-submodules H_{0}(R, ∗) = (H_{0}^{+}, H_{0}^{−}) ⊆ H(R, ∗) satisfying

x + x^{∗} ∈ H_{0}^{σ}, xH_{0}^{−σ}x^{∗} ⊆ H_{0}^{σ} for all x ∈ R^{σ} (σ = ±).

These are readily seen to be outer ideals of the Jordan triple system or pair H(R, ∗). For associative algebras, the presence of the unital hull behind the scenes leads us to demand that ample hermitian subspaces H_{0} = H_{0}(R, ∗) ⊆ H(R, ∗) satisfy an extra condition of closure under norms,

x + x^{∗}, xx^{∗} ∈ H_{0}, xH_{0}x^{∗} ⊆ H_{0} for all x ∈ R.

These are ample outer ideals of the Jordan algebra H(R, ∗) [21, 1.6, 1.7].

We will obtain a characterization of which (ample) outer ideals are ample hermitian subspaces. Moreover, this result is one of the key points of the paper since it will allow us to reduce the study of simplicity of outer ideals to that of ample hermitian subspaces. The work involves the notion of n-tad {x_{1}, …, x_{n}} = x_{1} ⋯ x_{n} + x_{n} ⋯ x_{1} of elements x_{1}, …, x_{n} of an associative system R.

0.8 Proposition. Let R be an associative system (algebra, pair, or triple system) with involution ∗, H = H(R, ∗).

(i) Any ample hermitian subspace H_{0} of R is an outer ideal of H (ample in the case of algebras) containing all appropriate n-tads of elements in H (odd n ≥ 1 for triple systems and pairs, all n ≥ 1 for algebras).

(ii) Conversely, if H_{0} is an ample hermitian subspace of R which generates R, i.e.,

(a) R = ∑ H_{0} ⋯ H_{0},

and K ⊆ H_{0} is a P -invariant submodule which contains all appropriate n-tads from H_{0}, i.e.,

(b) P_{H_{0}}K ⊆ K (also (b_{2}) H_{0}^{2} ⊆ K in algebras) , (c) {H_{0}, …, H_{0}} ⊆ K,

then K is an ample hermitian subspace of R.

Slightly more generally,

(ii’) If H_{0} is an ample hermitian subspace of R and K ⊆ H_{0} is a submodule which generates R via H_{0}, i.e.,

(a’) R = ∑ H_{0} ⋯ K ⋯ H_{0},

and it is P -invariant and eats appropriate n-tads from H_{0}, i.e.,

(b’) P_{H_{0}}K ⊆ K (also (b’_{2}) K^{2} ⊆ K in algebras) , (c’) {H_{0}, …, K, …, H_{0}} ⊆ K,

then K is an ample hermitian subspace of R.

Consequently, if H generates R, then the ample hermitian subspaces of R are precisely the outer ideals of H (ample in the case of algebras) containing all n-tads of elements in H, for all odd n ≥ 1 (all n ≥ 1 in the case of algebras).

Proof: For (i), H_{0} contains all n-tads since they are traces, {h_{1}, h_{2} …, h_{n}} = x + x^{∗} for x = h_{1}h_{2} ⋯ h_{n}, and it is outer since P_{h}H_{0} = hH_{0}h^{∗} ⊆ H_{0} by starred P -closure and {H, H, H_{0}} ⊆ H_{0} by 3-tad containment. In the case of algebras we also have outer ampleness H^{2} + 2H ⊆ H_{0} since 2h = h + h^{∗} is a trace and h^{2} = hh^{∗} is a norm, hence fall in H_{0} by algebra hermitian ampleness.

For (ii), by the generation hypothesis (a) we can write any x ∈ R as x = ∑ h_{1} ⋯ h_{n} for h_{i} ∈ H_{0}. Then every trace is a sum of n-tads, x+x^{∗} = ∑{h_{1}, …, h_{n}} and so falls in K by hypothesis (c), and K is starred P -closed by

xKx^{∗} = ∑ P_{h_{1}} ⋯ P_{h_{n}}K + ∑{h_{1}, …, h_{n}, K, h′_{m}, …, h′_{1}},

where the first sum falls in K by P -closure (b), and the second sum is in K by (n + m + 1)-tad containment (c). In the algebra case we also have closure under norms since

xx^{∗} = ∑ P_{h_{1}} ⋯ P_{h_{n−1}}h_{n}^{2} + ∑{h_{1}, …, h_{n}, h′_{m}, …, h′_{1}},

where the first sum falls in K by P -closure (b) and square-closure (b_{2}), while the second sum falls in K by (n + m)-tad containment (c).

For the refinement (ii’), we show the hypotheses (a’),(b’),(c’) imply (a),(b),(c) and hence hermitian ampleness of K. Clearly (a’) implies (a). Next, (c’) with (a’) implies (c) since all n-tads {H_{0}, …, H_{0}} are traces x + x^{∗}, and (a’) guarantees that all traces

x + x^{∗} = ∑{h_{1}, …, k, …, h_{n}} (for h_{i} ∈ H_{0}, k ∈ K)

fall in K by n-tad eating (c’). Similarly, in the case of algebras, the above with (b’_{2}) K^{2} ⊆ K implies (b_{2}) H_{0}^{2} ⊆ K: all H_{0}^{2} are norms, and all norms xx^{∗} fall in K since, by (a’), we can write x = ∑ h_{1} ⋯ h_{n}k for h_{i} ∈ H_{0}, k ∈ K (using n-tad eating (c’) to move the k to the far right), so that

xx^{∗} = ∑ P_{h_{1}} ⋯ P_{h_{n}}k^{2} + ∑{h_{1}, …, h_{n}, k, k′, h′_{m} …, h′_{1}}

falls in K by square closure (b’_{2}) K^{2} ⊆ K and P -closure (b’), and by (n + m + 1)-tad eating (c’).

The last assertion follows from (i) and (ii) for the case H_{0} = H.

0.9 The crucial question about outer ideals K ◁_{out} S in a simple system S is whether K = K^{3}: the heart Heart(K) = K^{3} turns out to be always simple, so K itself will be simple if and only if K = K^{3}. One easy case is when all elements are von Neumann regular. Given an associative or Jordan pair V, an element x ∈ V^{σ} is said to be von Neumann regular if x ∈ Q_{x}V^{−σ}, i.e., x = Q_{x}y (x = xyx in the associative case) for some y ∈ V^{−σ}. For associative or Jordan algebras or triple systems S, von Neumann regularity of x ∈ S is that of the element x in the pair V (S). A system S (associative or Jordan, algebra or triple system or pair) is called von Neumann regular if all its elements are von Neumann regular.

It is well known that von Neumann regularity is inherited by symmetric elements H(R, ∗), but it is also inherited by ample hermitian subspaces H_{0}(R, ∗) and by outer ideals.

0.10 Lemma. (i) If R is a von Neumann regular associative system (algebra, pair, or triple system) with involution ∗, then any ample hermitian subspace H_{0}(R, ∗) is von Neumann regular.

(ii) If S is a von Neumann regular Jordan system (algebra, pair, or triple system), then any outer ideal K ◁_{out} S is von Neumann regular.

Proof: (i) It is a general fact that any symmetric element x = x^{∗} of a starred P - invariant submodule H_{0} of R (not necessarily ample hermitian) which is von Neumann regular in R is already von Neumann regular in H_{0}: if x = xyx for some y ∈ R then x = xyx = xyx^{∗} = xy(xyx)^{∗} = xyxy^{∗}x = xzx for z = yxy^{∗} ∈ yH_{0}y^{∗} ⊆ H_{0} by starred P -invariance of H_{0}.

(ii) Denote here by P the Jordan operator (even for algebras or pairs). If x ∈ K has x = P_{x}y for some y ∈ S, then x = P_{P_{x}y}y = P_{x}P_{y}P_{x}y (by the Fundamental Formula (0.2)(iv)) = P_{x}P_{y}x, and P_{y}x ∈ K since K is an outer ideal in S.

0.11 Local algebras of Jordan and associative systems, introduced in [23], are one of the most useful tools to study primitivity of Jordan pairs and triple systems (cf. [2, 3, 4]). They are the way to connect the category of algebras with those of pairs and triple systems.

Given an associative pair R = (R^{+}, R^{−}) and a ∈ R^{−σ}, the Φ-module R^{σ} becomes an associative algebra, denoted R^{σ(a)} and called the a-homotope of R, with product

x ·_{a} y = xay,

for any x, y ∈ R^{σ}. The set

Ker_{R} a = Ker a = {x ∈ R^{σ} | axa = 0}

is an ideal of R^{σ(a)} and the quotient

R_{a}^{σ} = R^{σ(a)}/ Ker a

is an associative algebra called the local algebra of R at a.

We have analogous concepts for Jordan systems [5, 0.6]. Given a Jordan pair V = (V^{+}, V^{−}) and b ∈ V^{−σ}, the Φ-module V^{σ} becomes a Jordan algebra, denoted V^{σ(b)} and called the b-homotope of V, with product x^{(2,b)} = Q_{x}b, U_{x}^{(b)}y = Q_{x}Q_{b}y for any x, y ∈ V^{σ}. The set Ker_{V} b = Kerb = {x ∈ V^{σ} | Q_{b}x = Q_{b}Q_{x}b = 0} is an ideal of V^{σ(b)} and the quotient V_{b}^{σ} = V^{σ(b)}/Kerb is a Jordan algebra called the local algebra of V at b. When V is nondegenerate or special, Ker_{V} b = {x ∈ V^{σ} | Q_{b}x = 0}. Homotopes and local algebras of an associative or Jordan algebra or triple system S at an element b are simply those of the pair V (S). The above notions are compatible with the functors V ( ) and T ( ) (cf. [4, 0.5]).

1. Simple PI Systems

1.1 Recall the notion of centroid of an associative or Jordan algebra. Given an associative algebra R the centroid Γ(R) of R is defined as

Γ(R) = {γ ∈ End_{Φ}(R) | γL_{x} = L_{x}γ, γR_{x} = R_{x}γ, for all x ∈ R},

where L_{x} (resp., R_{x}) is the left (resp., right) multiplication by x on R. By [13, p. 107] Γ(R) is just the centralizer of R as a module over its multiplication algebra. If R is unital, then the centroid can be identified with the center of R (cf. [13, Prop. 2 on p. 108]). Moreover [13, Prop. 1 on p. 107], Γ(R) is commutative if R has zero left or right annihilator. When R is prime, Γ(R) is a domain acting on R without torsion, and one can consider the central closure Γ(R)^{−1}R of R, which is an algebra over the field of fractions Γ(R)^{−1}Γ(R). When R is simple, Γ(R) is a field [13, Th. 1 on p. 108], and R is automatically centrally closed.

For a Jordan algebra J, the centroid Γ(J) of J is defined as [19, p. 298]

Γ(J) = {γ ∈ End_{Φ}(J) | γU_{x} = U_{x}γ, U_{γx} = γ^{2}U_{x}, (γx)^{2} = γ^{2}x^{2},
V_{x}γ = γV_{x}, for all x ∈ J}.

By [19, Th. 2], for a Jordan algebra J with zero extreme radical (for example, if J is semiprime), Γ(J) is an associative commutative ring and J can be considered as a Jordan algebra over Γ(J). When J is prime, Γ(J) is a domain acting on J without torsion [22, 4.2; 20, 2.8] and one can consider the central closure Γ(J)^{−1}J of J, which is an algebra over the field of fractions Γ(J)^{−1}Γ(J). When J is simple, then Γ(J) is a field [19, Th. 3] and J is centrally closed.

1.2 Recall that an associative or Jordan algebra S is PI if it strictly satisfies an essential polynomial. In the associative case essential just means having a leading monomial with coefficient 1, while in the Jordan case essential means that the image of the Jordan polynomial in the free special Jordan algebra is essential in the associative sense (cf. [22, 7.2]). We say that a Jordan algebra is CPI if it strictly satisfies some Clifford polynomial identity (cf. [3, 22]).

An associative or Jordan triple system S is homotope-PI (HPI) if there is an essential algebra polynomial whose homotopization in a new variable is strictly satisfied by S [If f = f(x_{1}, …, x_{n}) is a polynomial in the free Jordan algebra F J_{alg}(X), its z-homotopization f(z; x_{1}, …, x_{n}) = f^{(z)}(x_{1}, …, x_{n}) is the element of the free Jordan triple system F J_{trip}(X ∪ {z}) obtained by evaluating f at x_{1}, …, x_{n} in the z-homotope F J_{trip}(X ∪ {z})^{(z)}] (see [4, Sect. 2]). An associative or Jordan pair V is HPI if T (V) is so. We say that a Jordan triple system is CHPI if it strictly satisfies some Clifford homotope polynomial identity (cf. [3]), and a Jordan pair V is CHPI if T (V) is so.

Clifford algebra polynomials are by definition essential polynomials. Thus, CPI Jordan algebras are PI, while CHPI Jordan pairs or triple systems are HPI.

1.3 Proposition. (i) Simple PI associative algebras are left and right primitive, unital, finite dimensional over their centroids, and von Neumann regular.

(ii) Simple PI Jordan algebras are primitive, unital, equaling their socles, and von Neumann regular.

Proof: (i) Let R be a simple PI associative algebra. Then R is prime, hence the central closure Γ(R)^{−1}R is simple, unital (hence two-sided primitive), finite dimensional over Γ(R)^{−1}Γ(R) by the non-unital version of [14, Th. 2 on p. 57], and von Neumann regular by [26, Cor. 2.11.21]. But the centroid Γ(R) of R is a field by (1.1), hence R = Γ(R)^{−1}R and Γ(R)^{−1}Γ(R) = Γ(R), which proves (i).

(ii) Let J be a simple PI Jordan algebra. By [6, 2.4(ii)], J is strongly prime, hence the central closure Γ(J)^{−1}J is simple, unital (hence primitive) by [10, Th. 1.1]. Moreover, Γ(J)^{−1}J is still PI, hence equals its socle by [10, Th. 5.2], and it is von Neumann regular by [17, Cor. 2 on p. 118]. But the centroid Γ(J) of J is a field by (1.1), hence J = Γ(J)^{−1}J, which proves (ii).

1.4 Remark: For a semiprime associative pair or triple system R, a nonzero element x is von Neumann regular if and only if the local algebra of R at x is unital: Assume that R is a triple system (the pair case is similar). If x = xyx, thenit is readily seen that y + Ker_{R} x is the unit element in R_{x}. Conversely, if y + Ker_{R} x is a left unit element in R_{x}, then yxa − a ∈ Ker_{R} x for any a ∈ R, so (xyx − x)R(xyx − x) = 0 [(xyx − x)a(xyx − x) = (x(yxa − a))(xyx − x) = 0], which implies xyx = x by semiprimeness of R.

Using local algebras we can obtain a pair and triple system version of (1.3). By [24], we just need to assume the condition of being PI on one of the local algebras.

1.5 Lemma. Let S be a simple associative or Jordan pair or triple system. If there exists a nonzero element in S at which the local algebra of S is PI, then all the local algebras of S are PI.

Proof: By [24, 1.6] in the associative case and [24, 5.4] in the Jordan case (since S is nondegenerate by [6, 2.4(ii)]), the set PI(S) of elements at which the local algebras are PI is an ideal of V. By simplicity, this ideal is the whole of S.

1.6 Proposition. (i) Simple associative pairs and triple systems having a PI local algebra at a nonzero element (for example, when they are HPI ) are left and right primitive and von Neumann regular.

(ii) Simple Jordan pairs and triple systems having a PI local algebra at a nonzero element (for example, when they are HPI ) are primitive, equaling their socles, hence they are von Neumann regular.

Proof: (i) Let R be a simple associative pair or triple system having a PI local algebra at a nonzero element. By (1.5), all local algebras of R are PI. On the other hand, there exists some nonzero a in R such that the local algebra of R at a is simple (see [11, 5.2(iii)] and [6, 2.6]) (indeed, this holds for all nonzero a in the pair case). This local algebra turns out to be unital and two-sided primitive by (1.3)(i), hence R is two-sided primitive by [4, 1.8].

In the case of pairs, the previous argument applies to all local algebras, hence all of them are unital. For a simple triple system, there may be nonsimple local algebras. However, they are either simple or the direct sum of two simple algebras (cf. [6, 2.6]), both of them necessarily PI in our case, hence unital as above. Thus, all local algebras of R are unital and R is von Neumann regular by (1.4).

(ii) Let S be a simple Jordan pair or triple system having a PI local algebra at a nonzero element. By (1.5) all local algebras of S are PI. By [6, 2.5, 2.6] there exists some local algebra of S which is simple, hence primitive equaling its socle by (1.3)(ii). Thus, S is primitive by [4, 3.8, 3.9] since it is strongly prime by [6, 2.4(ii)], and the socle of S is nonzero by [2, Th. 2.1]. By simplicity of S, S equals its socle and is von Neumann regular by [17, Th. 1, Cor. 2 on p. 118].

2. Simplicity of outer ideals in Jordan algebras

We recall the following result about simplicity of ample outer ideals of unital Jordan algebras.

2.1 Proposition [21, Th. 5.1]. If J is a unital simple Jordan algebra, then any ample outer ideal K ◁_{out} J is also simple and unital.

In the following lemmas we will show that the study of outer ideals of simple Jordan algebras can be essentially reduced to the study of ample outer ideals.

2.2 Lemma. Let J be a Jordan algebra.

(i) If K is a squared outer ideal of J, then J ∘ U_{K} Ĵ ⊆ K.

(ii) If K, L are (resp., squared) outer ideals of J, so are K + L, K ∩ L and U_{K}L. Moreover, in the squared case, (U_{K}L)^{2} ⊆ K ∩ U_{K}L.

(iii) An outer ideal K is an ideal if and only if U_{k_{i}} Ĵ ⊆ K for all k_{i} in some spanning set {k_{i}} for K.

(iv) If K is any outer ideal of J, then U_{K}J is an ideal of J.

Proof: (i) follows from (0.2)(v). (ii) The assertions for the sum and intersection are clear (note for squaredness that always K ∘ L ⊆ K ∩ L). To prove that U_{K}L is a (resp., squared) outer ideal of J whenever K, L are, we have to show that J ∘ U_{K}L + U_{J}U_{K}L ⊆ U_{K}L (resp., and (U_{K}L)^{2} ⊆ U_{K}L).

(I) J ∘ U_{K}L ⊆ U_{K}L: take x ∈ J, a ∈ K, b ∈ L, hence x ∘ U_{a}b = {x ∘ a, b, a} − U_{a}(x ∘ b) (by (0.2)(ii)) ∈ {K, L, K} + U_{K}L = U_{K}L. If K is squared, then also J ∘ U_{K}L ⊆ K ∩ U_{K}L by (i).

(II) U_{J}U_{K}L ⊆ U_{K}L: take x ∈ J, a ∈ K, b ∈ L, hence U_{x}U_{a}b = U_{x∘a}b + U_{U_{x}a,a}b − U_{a}U_{x}b−x∘U_{a}(x∘b) (by (0.2)(vi)) ∈ U_{K}L+{K, L, K}+U_{K}L+J ∘U_{K}L ⊆ U_{K}L, by (I).

(III) In algebras (U_{K}L)^{2} is spanned by all (U_{a}b)^{2} and U_{a}b∘U_{a′}b′ for a, a′ ∈ K, b, b′ ∈ L where (U_{a}b)^{2} = U_{a}U_{b}a^{2} (by (0.2)(iv)) lies in U_{J}U_{J}K^{2} ⊆ K when K is squared outer, and it also equals U_{a}((a ∘ b)^{2} − U_{a}b^{2} − b ∘ U_{a}b) (by (0.2)(vii)) ∈ U_{a}L^{2} − U_{a^{2}}L^{2} − U_{a}(L ∘ J) ⊆ U_{K}L when K^{2} ⊆ K, L^{2} ⊆ L, and U_{a}b ∘ U_{a′}b′ ∈ J ∘ U_{K}L ⊆

K ∩ U_{K}L by (I).

(iii) Any k ∈ K has k = ∑ α_{i}k_{i}, so U_{k}Ĵ ⊆ ∑ α_{i}^{2}U_{k_{i}} Ĵ+ ∑ α_{i}α_{j}{k_{i}, Ĵ, k_{j}} where U_{k_{i}} Ĵ ⊆ K by hypothesis and {k_{i}, Ĵ, k_{j}} ⊆ {J, Ĵ, K} ⊆ K, so U_{K} Ĵ ⊆ K and K is inner.

(iv) U_{K}J is outer by (ii), and it is an ideal by (iii) since for spanning elements k = U_{a}x (a ∈ K, x ∈ J) we have U_{k}Ĵ = U_{a}U_{x}U_{a}Ĵ (by (0.2)(iv)) ⊆ U_{a}J ⊆ U_{K}J.

2.3 Corollary. (i) If K is a nonzero outer ideal of a simple Jordan algebra J, then J = U_{K}J.

(ii) If K is a squared outer ideal of a Jordan algebra J such that J = U_{K}J, then K is ample: 2J + J^{2} ⊆ K.

(iii) If K is a nonzero squared outer ideal of a simple Jordan algebra J, then K and U_{K}K are ample outer ideals of J and J = U_{K}J = U_{U_{K}K}J.

(iv) If J is a unital simple Jordan algebra, then so is any nonzero squared outer ideal K of J.

Proof: (i) Simplicity of J implies nondegeneracy [6, 2.4(ii)], hence U_{K}J ≠ 0 is an ideal of J by (2.2)(iv). (ii) 2J = 2U_{K}J = {K, J, K} ⊆ K by outerness of K, and J^{2} = (U_{K}J)^{2} ⊆ K by (2.2)(ii). (iii) Since J is simple it is nondegenerate by [6, 2.4(ii)], hence K is nondegenerate by [21, 2.1], which implies U_{K}K ≠ 0, and we apply (2.2)(ii) and (i),(ii) above. (iv) results by putting together (2.1) and (iii).

2.4 Remark: If H_{0} is an ample hermitian subspace of an associative system (algebra, pair, or triple system) R with involution ∗, we may assume that R is a ∗-tight cover of H_{0}, i.e. R is generated by H_{0} and R is a ∗-tight extension in the sense that any nonzero ∗-ideal of R has nonzero intersection with H_{0}. We certainly can always replace R by the cover R̃, the subsystem of R generated by H_{0} (H_{0} is even more ample in R̃), and then tighten by replacing R̃ with R̃/I, where I is a maximal ∗-ideal of R̃ not hitting H_{0}. Tight covers inherit simplicity: if H_{0} is simple then R is ∗-simple.

2.5 Proposition. Let H_{0} = H_{0}(R, ∗) be a simple ample hermitian subspace of an associative system (algebra, pair, or triple system) R with involution ∗ and assume that R is a ∗-tight cover of H_{0}, and let K be a nonzero (squared in the algebra case) outer ideal of H_{0}. If H_{0} is non-CHPI (non-CPI in the algebra case) (for example, if it is a non-HPI pair or triple system or a non-PI algebra), then K is also an ample hermitian subspace of R, and K^{3} = Heart(K) is simple. If, in addition, R is not simple, then K = H_{0} ≅ S^{(+)} for a simple associative system S, hence K is simple.

Proof: \underline{Case of a triple system:} To show that K is an ample hermitian subspace of R, we just need to check the hypothesis of (0.8)(ii’).

The ideal of R generated by K is a nonzero ∗-ideal of R, and we noted above that R is ∗-simple, so R = id_{R}(K). Since the cover R is generated by H_{0}, we have

(a’) R = id_{R}(K) = ∑ H_{0} ⋯ K ⋯ H_{0}.

Now notice that H_{0} is simple and non-CHPI, so it coincides with HT_{n}(H_{0}) for all odd n ≥ 5 (see [8, 1.4]), i.e., its elements “eat” as much as we like. We claim that

(c’) {H_{0}, …, K, …, H_{0}} ⊆ K.

Indeed, by the eating property of the elements of H_{0} [8, 1.2], any such n-tad can be expressed as a sum of terms of the form {p_{1}, p_{2}, p_{3}}, where p_{1}, p_{2}, p_{3} are Jordan polynomials in H_{0}, and an element from K appears linearly in one of them. Because of this linearity, the action on K is necessarily outer, which leads to the fact that one of the three polynomials p_{j} ∈ K. Therefore {p_{1}, p_{2}, p_{3}} ∈ K, establishing (c’).

Furthermore, since K is an outer ideal of H_{0} it also satisfies (b’) of (0.8)(ii’), so, by that proposition, K is an ample hermitian subspace of R.

Now, K^{3} = Heart(K) is simple by [7, 3.15(ii)], and if R is not simple, R ≅ S⊕S^{op} for a simple S with exchange involution, hence H_{0} = H_{0}(R, ∗) ⊆ H(R, ∗) = {s + s^{∗} | s ∈ S} ⊆ K and K = H_{0} ≅ S^{(+)} is simple by [7, 1.7(ii)].

\underline{Case of an algebra:} The proof in the triple system situation holds for algebras with suitable changes. Notice that for such an algebra H_{0} we have H_{0} = H_{n}(H_{0}) for all n by simplicity and [8, 1.1, 1.3], so that we have (c’) of (0.8)(ii)’ too. Since here K is an squared outer ideal of H_{0}, we also have (b’) and (b’_{2}) of (0.8)(ii’), hence K is an ample hermitian subspace of R. The rest of assertions follow in the same way as for a triple system, but applying [7, 2.7(ii)] and [18, Th. 4].

\underline{Case of a pair:} The result for pairs is obtained from the triple system result by using the functor T ( ). We first notice that T (H_{0}) is simple by (0.4)(i), and it is a non-CHPI ample hermitian subspace of the polarized triple system T (R). Moreover, since R is ∗-simple by ∗-tightness over H_{0}, which is simple, T (R) is also ∗-simple by (0.4)(i), and it is a ∗-tight cover of T (H_{0}).

On the other hand, T (K) is readily seen to be an outer ideal of T (H_{0}) hence, by the triple system case, it is an ample hermitian subspace of T (R) and, at the “triple level”, T (K)^{3} = Heart(T (K)) is simple. As an immediate consequence, K is an ample hermitian subspace of R. Notice that H_{0} is nondegenerate by [6, 2.4(ii)], so that K is nondegenerate by [5, 2.1], and we can use [8, 3.7(ii)] to obtain Heart(T (K)) = T (Heart(K)). On the other hand, T (K^{3}) = T (K)^{3} = Heart(T (K)), so that we can “cancel” T to get K^{3} = Heart(K) and it is simple by (0.4)(i). Finally, if R is not simple we can argue as in the triple system case.

2.6 Theorem. If K is a nonzero squared outer ideal of a simple Jordan algebra J, then K^{3} is simple, and either

(i) K = K^{3} itself is simple, or

(ii) K is an ample hermitian subspace of some simple non-unital and non-PI associative algebra R with involution of characteristic 2 (2R = 0).

Proof: A Jordan algebra J is either (1) exceptional, (2) special unital, or (3) special non-unital. From the classification theorem [22, 15.5], simple algebras of type (1) are necessarily unital, while those of type (3) are of the form J = H_{0}(R, ∗) for a ∗-simple associative algebra R (since R can be assumed to be a ∗-tight cover of J by (2.4)). By (2.3)(iv), K is simple as soon as J is unital, hence, the only possible case in which K might not be simple is (3) for a non-unital R. Moreover, if R or J were PI, then they would be unital by (1.3) (notice that if R is not simple, it is the direct sum of two simple algebras by ∗-simplicity). Thus, we may assume that R and J are both non-PI.

In this last case, K is an ample hermitian subspace of R, K^{3} is simple, and K is simple if R is not simple by (2.5). Finally, if 2J ≠ 0 then the centroid Γ(J) is a field of characteristic not two, which implies 1/2 ∈ Γ(J), and outer ideals are just ideals, getting K = J by simplicity of J. Thus the only possible case where K might not be simple is (ii) since 2J = 0 implies 2R = 0 (R is generated by J).

Notice that in all cases K^{3} is simple (when K is simple, K^{3} is a nonzero ideal of K, hence K^{3} = K is simple).

Now, we can extend (2.3)(iv) to von Neumann regular Jordan algebras.

2.7 Corollary. Let J be a simple von Neumann regular Jordan algebra and let K be a nonzero squared outer ideal of J. Then K = K^{3} is simple.

Proof: By (0.10)(ii), K is von Neumann regular, hence K = K^{3}, where K^{3} is simple by (2.6).

3. Simplicity of outer ideals in Jordan pairs and triple systems

In view of (2.5), our next goal is to study outer ideals of simple Jordan systems which are CHPI, or more generally HPI.

3.1 Proposition. If V is a simple Jordan pair having a PI local algebra at a nonzero element (for example, when V is HPI ), then any nonzero outer ideal K◁_{out}V is simple.

Proof: By (1.5), all local algebras of V are PI. Let 0 ≠ x ∈ K^{−σ}, σ = ±. It is easy to check that

(I) K_{x}^{σ} is isomorphic to the squared outer ideal (K^{σ} + Ker x)/ Ker x of V_{x}^{σ}.

Moreover, K is nondegenerate by [5, 2.1] since V is nondegenerate by [6, 2.4(ii)], hence

(II) K_{x}^{σ} ≠ 0.

Using [6, 2.5], V_{x}^{σ} is simple. Moreover, it is also PI, hence (1.3)(ii) yields

(III) V_{x}^{σ} is simple and unital.

Now, (I–III) imply K_{x}^{σ} is simple by (2.3)(iv). Since this holds for any 0 ≠ x ∈ K^{−σ}, σ = ±, K^{3} is simple by [6, 4.8]. But V is von Neumann regular by (1.6)(ii), hence K is von Neumann regular by (0.10)(ii), and K = K^{3}, which implies that K is simple.

Let us now obtain the analogue of this result for Jordan triple systems using the functor V ( ).

3.2 Proposition. If T is a simple Jordan triple system having a PI local algebra at a nonzero element (for example, when T is HPI ), then any nonzero outer ideal K ◁_{out} T is simple.

Proof: By (1.5), all local algebras of T are PI. Let us consider V = V (T), the double pair associated to the triple system T. Then we have one of the following two possibilities (cf. [6, 2.6]):

(I) V is simple. In this case V (K) is an outer ideal of V and V is a simple pair having a PI local algebra at a nonzero element, hence V (K) is simple by (3.1) and therefore K is also simple by (0.4)(ii).

(II) V = W ⊕ W^{op} for a simple Jordan pair W and T = T (W) = W^{+} ⊕ W^{−} is a polarized triple system. Consider π^{σ}(K) := {x ∈ W^{σ} | ∃y ∈ W^{−σ} with x + y ∈ K}, σ = ±, and set \overline{K} = π^{+}(K) + π^{−}(K) and \underline{K} = (K ∩ W^{+}) + (K ∩ W^{−}). We claim that \underline{K} ⊆ K ⊆ \overline{K} are outer ideals of T. Indeed,

(P_{T} + L_{T,T})\overline{K} ⊆ \underline{K}

since

P_{T} \overline{K} = P_{T} (π^{+}(K) + π^{−}(K))
= P_{T} (π^{+}(K)) + P_{T} (π^{−}(K))
= P_{W^{−}}(π^{+}(K)) + P_{W^{+}}(π^{−}(K))
= P_{W^{−}}K + P_{W^{+}}K
⊆ (K ∩ W^{−}) + (K ∩ W^{+}) = \underline{K}

by outerness of K, and similarly

L_{T,T} \overline{K} = L_{W^{+},W^{−}}(π^{+}(K)) + L_{W^{−},W^{+}}(π^{−}(K))
= L_{W^{+},W^{−}}K + L_{W^{−},W^{+}}K
⊆ (K ∩ W^{+}) + (K ∩ W^{−}) = \underline{K}.

Then also (π^{+}(K), π^{−}(K)) is a nonzero outer ideal of W and since this latter is simple and has a PI local algebra at a nonzero element, we get that (π^{+}(K), π^{−}(K)) is simple by (3.1), hence also \overline{K} = π^{+}(K) ⊕ π^{−}(K) is simple as a triple system by (0.4)(i). But K is a triple ideal of \overline{K}: it is outer in T, and inner in \overline{K} since P_{K} \overline{K} ⊆ P_{T} \overline{K} ⊆ \underline{K} ⊆ K.

We have shown that K ≠ 0 is an ideal of \overline{K} and that \overline{K} is simple, so K = \overline{K} is simple.

We can now extend (2.6) to pairs and triple systems.

3.3 Theorem. Let S be a simple Jordan pair or triple system and let K be a nonzero outer ideal of S. Then K^{3} is simple, and either

(i) K itself is simple.

(ii) K is an ample hermitian subspace of some simple, non von-Neumann regular associative system R with involution, without nonzero PI local algebras (in particular, non-HPI ) of characteristic 2 (2R = 0).

Proof: Let S be a simple Jordan pair or triple system. If S is exceptional, then it is outer simple [27, Th. 6], hence K = S is simple. If S is special and CHPI, then it has only simple outer ideals by (3.1) and (3.2) since it is HPI.

On the other hand, if S is special and non-CHPI, then S = H_{0}(R, ∗) is an ample hermitian subspace of an associative system R with involution ∗ [1, 4.4, 5.5], which can be assumed to be a ∗-tight cover of S by (2.4). Therefore, by (2.5), K is also an ample hermitian subspace of R with K^{3} simple, and K itself is simple when R is not simple.

If R is simple, having a PI local algebra at a nonzero element, then it is von Neumann regular by (1.6)(i), and, once it is von Neumann regular, both S and K are von Neumann regular by (0.10), which implies that K = K^{3} is simple. If 2S ≠ 0, then the centroid Γ(S) is a field of characteristic not two, 1/2 ∈ Γ(S), and outer ideals are just ideals, which gives K = S by simplicity of S. When 2S = 0, 2R = 0 too since R is generated by S.

Replacing (2.6) by (3.3) in the proof of (2.7) gives a pair and triple version of this result.

3.4 Corollary. If S is a simple von Neumann regular Jordan pair or triple system, then any nonzero outer ideal K ◁_{out} S is simple.

4. Outer Ideals in Jordan Pairs Covered by Ortho-Collinear Grids

Some particular types of Jordan pairs allow a direct study of outer ideals, which shows how their simplicity and outer simplicity can be given in terms of their coordinate algebras.

4.1 Proposition. Let A be a unital associative algebra and let V = 𝕄_{p,q}(A) = (Mat_{p,q}(A), Mat_{q,p}(A))^{(+)} denote the matrix pair over A for 3 ≤ p + q ≤ ∞. Then any outer ideal K of V has the form 𝕄_{p,q}(I) where I is an ideal of A, hence K is actually an ideal. Therefore, V is outer-simple if and only if V is simple, if and only if A is simple.

Proof: Let us denote by E_{ij}^{σ} the elementary matrix of V^{σ} with zero entries in every position but a 1 in the position ij. They form a grid of idempotents that covers the pair V (see [25]). Define

I^{σ} := {x ∈ A | ∃i, j with xE_{ij}^{σ} ∈ K^{σ}}, I := I^{+} ∪ I^{−}.

We are going to show that K = 𝕄_{p,q}(I).

(I) If ∑_{k,l} x_{kl}E_{kl}^{σ} ∈ K^{σ}, then x_{ij}E_{ij}^{σ} ∈ K^{σ} for every i, j. Thus x_{ij} ∈ I^{σ} ⊆ I and K ⊆ 𝕄_{p,q}(I). Indeed,

Q_{E_{ij}^{σ}}Q_{E_{ji}^{−σ}}(∑_{k,l} x_{kl}E_{kl}^{σ}) = x_{ij}E_{ij}^{σ} ∈ K^{σ}

for every i, j.

On the other hand, (II) If xE_{ij}^{σ} ∈ K^{σ}, then xE_{ji}^{−σ} ∈ K^{−σ}, so I = I^{−σ} = I^{σ}. Indeed,

Q_{E_{ji}^{−σ}}(xE_{ij}^{σ}) = xE_{ji}^{−σ} ∈ K^{−σ}.

(III) If xE_{ij}^{σ} ∈ K^{σ} then xE_{ip}^{σ} ∈ K^{σ} for any σ-column index p, and xE_{rj}^{σ} ∈ K^{σ} for any σ-row index r. Indeed, for any p ≠ j, r ≠ i,

{xE_{ij}^{σ}, E_{ji}^{−σ}, E_{ip}^{σ}} = xE_{ip}^{σ} ∈ K^{σ}, and {E_{rj}^{σ}, E_{ji}^{−σ}, xE_{ij}^{σ}} = xE_{rj}^{σ} ∈ K^{σ}.

Thus I = {x ∈ A | xE_{11}^{+} ∈ K^{+}} = {x ∈ A | xE_{11}^{−} ∈ K^{−}} and x ∈ I implies all xE_{ij}^{σ} ∈ K^{σ}, thus 𝕄_{p,q}(I) ⊆ K, and we have shown that K = 𝕄_{p,q}(I). Moreover, I is an ideal of A: Let x ∈ I and y ∈ A and suppose, for example, that q ≥ 2. Then xE_{11}^{+} ∈ K^{+}, yE_{11}^{−} ∈ V^{−} and E_{12}^{+} ∈ V^{+}, hence

{xE_{11}^{+}, yE_{11}^{−}, E_{12}^{+}} = xyE_{12}^{+} ∈ K^{+},

i.e., xy ∈ I. And similarly, AI ⊆ I.

This shows that any outer ideal of V has the form 𝕄_{p,q}(I) for an ideal I of A, and these are ideals of 𝕄_{p,q}(A) (see [12, 3.8]), so outer ideals are indeed ideals. Moreover it is clear that 𝕄_{p,q}(A) is (outer) simple if and only if A is simple.

4.2 Proposition. Let A be a unital commutative associative algebra and let V = 𝔸_{q}(A) = (Alt_{q}(A), Alt_{q}(A)) denote the alternating matrix pair over A for 3 ≤ q ≤ ∞. Then any outer ideal K of V has the form 𝔸_{q}(I) where I is an ideal of A, hence K is actually an ideal. Therefore, V is outer-simple if and only if V is simple, if and only if A is simple, if and only if A is a field.

Proof: Let us denote by [ij]^{σ} = 1[ij]^{σ} = E_{ij}^{σ} −E_{ji}^{σ} (note that [ii]^{σ} = 0), σ = ±. They form a grid of idempotents which cover the pair V (see [25]). As in the above proof of (4.1) we again define

I^{σ} := {x ∈ A | ∃i, j with x[ij]^{σ} ∈ K^{σ}},

and we are going to show that K = 𝔸_{q}(I) for I = I^{+} = I^{−}.

(I) If ∑_{k<l} x_{kl}[kl]^{σ} ∈ K^{σ}, then x_{ij}[ij]^{σ} ∈ K^{σ} for every i < j (note that x[ij]^{σ} = −x[ji]^{σ}). Thus K^{σ} ⊆ Alt_{q}(I^{σ}). Indeed,

Q_{[ij]^{σ}}Q_{[ij]^{−σ}}(∑_{k<l} x_{kl}[kl]^{σ}) = x_{ij}[ij]^{σ} ∈ K^{σ}

for every i < j.

(II) If x[ij]^{σ} ∈ K^{σ}, then x[ij]^{−σ} ∈ K^{−σ}, so I^{+} = I^{−}. Indeed,

Q_{[ij]^{−σ}}x[ij]^{σ} = −x[ij]^{−σ} ∈ K^{−σ},

hence also x[ij]^{−σ} ∈ K^{−σ}.

(III) If x[ij]^{σ} ∈ K^{σ} (i ≠ j) then x[ip]^{σ} ∈ K^{σ} for any p ≠ i, j, and x[rj]^{σ} ∈ K^{σ} for any r ≠ i, j. Indeed,

{x[ij]^{σ}, [ji]^{−σ}, [ip]^{σ}} = x[ip]^{σ} ∈ K^{σ} and {[rj]^{σ}, [ji]^{−σ}, x[ij]^{σ}} = x[rj]^{σ} ∈ K^{σ}.

Thus I = {x ∈ A | x[12]^{+} ∈ K^{+}} = {x ∈ A | x[12]^{−} ∈ K^{−}} and x ∈ I implies all x[ij]^{σ} ∈ K^{σ}, hence 𝔸_{q}(I) ⊆ K.

(I) and (III) show K = 𝔸_{q}(I). Moreover, I is an ideal of A since for any x ∈ I and any y ∈ A, x[12]^{+} ∈ K^{+}, y[23]^{−} ∈ V^{−} and [32]^{+} ∈ V^{+} so

{x[12]^{+}, y[23]^{−}, [32]^{+}} = xy[12]^{+} ∈ K^{+},

hence xy ∈ I. Thus IA ⊆ I, similarly (or by commutativity) AI ⊆ I, and I is an ideal.

This shows that any outer ideal of V has the form 𝔸_{q}(I) for an ideal I of A, and these are ideals of 𝔸_{q}(A) (see [12, 3.7(1)]), so outer ideals are indeed ideals. Moreover it is clear that 𝔸_{q}(A) is (outer) simple if and only if A is simple.

Pairs considered in (4.1) and (4.2) are particular cases of pairs covered by connected ortho-collinear grids of size ≥ 2 (cf. [25]). In this more general class of Jordan pairs the equality between outer ideals and ideals also holds, though we will not obtain a precise description of their form in terms of the coordinate algebras.

4.3 Proposition. Let V be a Jordan pair covered by a connected ortho-collinear grid (not necessarily finite) of size ≥ 2. Then every outer ideal is an ideal. Hence V is simple if and only if V is outer simple.

Proof: Let G be an ortho-collinear grid (not necessarily finite) covering V. Hence V = ⊕_{g∈G}V_{g} where V_{g} is the Peirce space of V with respect to G that contains g ∈ G. Since by assumption on G two distinct idempotents of G are either orthogonal or collinear, V_{g′} ⊆ V_{2}(g′) ∩ V_{i}(g) for i = 0 [resp., 1] in case g and g′ are orthogonal [respectively, collinear]. Thus, as a consequence of Peirce multiplication rules [25, 1.3], we have

Q(V_{g}^{σ})V_{g′}^{−σ} = \{ \begin{matrix}V_{g}^{σ} & if g = g′, \\ 0 & if g ≠ g′ \end{matrix}  (1)

for σ = ±. Let now K be an outer ideal of V. For any k^{σ} ∈ K^{σ}, k^{σ} = ∑_{g∈G} k_{g}^{σ}, where k_{g}^{σ} ∈ V_{g}^{σ}. Since V_{g} ⊆ V_{2}(g), it follows from (1) that k_{g}^{σ} = Q(g^{σ})Q(g^{−σ})k^{σ} ∈ K^{σ} by outerness of K. Hence

K^{σ} = ⨁_{g∈G} K_{g}^{σ},  for K_{g}^{σ} = K^{σ} ∩ V_{g}^{σ}  (2)

for σ = ±. To show that K is an ideal of V we need to verify that Q(K^{σ})V^{−σ} ⊆ K^{σ} (σ = ±). But, since {V^{σ}, V^{−σ}, K^{σ}} ⊆ K^{σ} by outerness, it suffices in view of (1) and (2) to prove that

Q(K_{g}^{σ})V_{g}^{−σ} ⊆ K^{σ}

for every g ∈ G, σ = ±. Since G is connected and ortho-collinear of size ≥ 2 there exists an idempotent h ∈ G that is collinear to g. Then V_{g} ⊆ V_{2}(g)∩V_{1}(h) and hence, for any k^{σ} ∈ K_{g}^{σ}, y^{−σ} ∈ V_{g}^{−σ}

Q_{k^{σ}}(y^{−σ}) = Q_{k^{σ}}({h^{−σ}, h^{σ}, y^{−σ}}) (by [25, 1.3]) = Q_{k^{σ}}D_{y^{−σ},h^{σ}}(h^{−σ}) =

(Q_{{h^{σ},y^{−σ},k^{σ}},k^{σ}} − D_{h^{σ},y^{−σ}}Q_{k^{σ}})(h^{−σ}) (by [16, JP12]) =
{{h^{σ}, y^{−σ}, k^{σ}}, h^{−σ}, k^{σ}} − 0 (by (1)) ∈ {V^{σ}, V^{−σ}, K^{σ}} ⊆ K^{σ}.

4.4 Remark: Apart from the pairs considered in (4.1) and (4.2), the following pairs are covered by connected ortho-collinear grids of size ≥ 2 (so that their outer ideals are ideals by (4.3)):

(i) 𝕄_{12}(A) for A a unital alternative algebra, including the various “16-dimen-sional” exceptional Jordan pairs (the covering grid here is a rectangular grid of size 2 or a bi-Caley grid).

(ii) Jordan pairs associated to a quadratic form which is an orthogonal sum of hyperbolic planes (covered by an even quadratic form grid).

(iii) The Jordan pair associated to an exceptional algebra (after an extension, this is covered by an Albert grid).

Acknowledgments

We thank the referee for allowing us to include his or her result (Proposition 4.3) in our paper, as well as for his or her many valuable comments and suggestions.

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