PRIMITIVE ALGEBRAS WITH PI PRIMITIZERS ^{1}

José A. Anquela, Esther García

Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain

Abstract: We show that a primitive (resp. ∗-primitive) associative algebra having a nonzero PI primitizer (resp. ∗-primitizer) is necessarily PI. Similarly, a primitive Jordan algebra having a PI primitizer with nonzero square is PI.

Introduction

Primitizers satisfying a polynomial identity appear naturally in the solution [\underline{3}, \underline{4}] of the problem of maximality of maximal modular inner ideals in Jordan algebras [\underline{8}]. In [\underline{4}] it is shown that a special primitive Jordan algebra having a PI primitizer is necessarily simple and unital. As an immediate consequence of the description of primitive Jordan algebras [\underline{5}], the above result holds for not-necessarily special Jordan algebras.

We will consider primitive Jordan and associative algebras having a PI primitizer. The aim of this paper is studying how far these algebras are from being “globally” PI. For associative algebras we will show that just assuming that the PI primitizer is nonzero is enough to guarantee that the whole algebra is PI, while for Jordan algebras we will also need that the PI primitizer has nonzero square to obtain a similar result.

The paper is divided into three sections. After a preliminary section containing the first definitions and known results, we show that the associative arguments used in [\underline{4}] to study special Jordan algebras are valid in the even more natural setting of an associative algebra R having a PI primitizer K, obtaining that R is simple and unital. From that fact, representation arguments are used to obtain that R is PI if K is nonzero. In most of the results of this section we consider simultaneously associative algebras with and without involution since the proofs can be adapted from one case to the other with very little effort.

1 Partially supported by the II Plan Regional de Investigación del Principado de Asturias FICYT-PB-PGI98-05 and the DGES PB97-1069-C02-02.

In the second section the results concerning Jordan algebras are obtained from their associative versions by using the classification of simple Jordan algebras [\underline{11}] and the results of Amitsur involving identities in rings with involution [\underline{1}].

The last section is devoted to checking the necessity of the hypothesis in some of the previously established results, as well as to compare them with some related results that appeared in the literature. Among other things, we study the possibility (indeed the lack of it) of obtaining information on the dimension of the algebra from the degree of a polynomial identity satisfied by a primitizer (cf. [\underline{7}, Kaplansky’s Theorem 6.3.1]). We also compare our results with those by Montaner in his study of Jordan algebras with PI local algebras [\underline{13}].

0. Preliminaries

0.1 We will deal with associative and Jordan algebras over an arbitrary ring of scalars Φ. The reader is referred to [\underline{5}, \underline{11}, \underline{14}] for basic results, notation and terminology.

0.2 We will consider polynomial identities in associative and Jordan algebras. An associative or Jordan algebra satisfying a polynomial identity will be called PI. Basic facts concerning PI associative algebras can be found in [\underline{7}] and [\underline{14}, vol. II], while polynomial identities in Jordan algebras are clearly defined and extensively used in [\underline{11}].

0.3 Recall the notion of modularity for one-sided ideals of associative algebras: A left (resp. right) ideal K of an associative algebra is called modular if there exists an element e ∈ R, called a modulus for K, such that x − xe ∈ K (resp. x − ex ∈ K) for any x ∈ R.

Note that we can shift any modulus by an element of K: if e is a modulus for K, so is e′ = e + k for any k ∈ K.

0.4 If R has an involution then any modulus e for a left ideal K can be replaced by a symmetric modulus ē = e+e^{∗}−e^{∗}e (analogously ē = e+e^{∗}−ee^{∗} for a right ideal) since k = e^{∗} − e^{∗}e ∈ K and ē = e + k is a translate of e by k ∈ K, hence is again a modulus by (0.3).

0.5 An associative algebra R is called left (resp. right) primitive if one of the following equivalent [\underline{7}, Th. 2.1.1, 2.1.2] conditions holds:

(i) R possesses an irreducible faithful left (resp. right) module.

(ii) R is a dense ring of left (resp. right) linear transformations of a vector space over a division algebra.

(iii) There exists a maximal modular left (resp. right) ideal K of R with zero core (i.e., K does not contain nonzero ideals of R).

By arguing as in [\underline{5}, 0.6] we also have the following equivalent assertion:

(iv) There exists a left (resp. right) primitizer of R, i.e., a left (resp. right) modular ideal K of R such that it is proper (K ≠ R) and complements all nonzero ideals of R (I + K = R for any nonzero ideal I of R).

We will also use the following equivalent condition:

(v) There exists a proper left (resp. right) modular ideal K of R such that every nonzero ideal of R contains a modulus for K.

Indeed, if K complements an ideal I and e is a modulus for K, then e = y +k for some y ∈ I and k ∈ K, and y = e − k ∈ I is a modulus for K; conversely, if the ideal I contains a modulus e for K then I + K is modular and contains its modulus e, so it must be all of R.

0.6 There is not a suitable notion of one-sided representation for associative algebras with involution which provides a suitable density theorem. To get the notion of ∗-primitivity, we just replace in (0.5)(iii) or (iv) ideals by ∗-ideals. Thus an associative algebra R with involution ∗ is called left (resp. right) ∗-primitive if one of the following equivalent [\underline{5}, 4.3] conditions holds:

(i) There exists a maximal modular left (resp. right) ideal K of R with zero ∗-core (i.e., K does not contain nonzero ∗-ideals of R).

(ii) There exists a left (resp. right) ∗-primitizer of R, i.e., a left (resp. right) modular ideal K of R such that it is proper (K ≠ R) and complements all nonzero ∗-ideals of R (I + K = R for any nonzero ∗-ideal I of R).

Since an involution is an antiautomorphism, every algebra R with involution is isomorphic to its opposite R^{op}, and the notions of left ∗-primitivity and right ∗-primitivity are equivalent.

0.7 It can be shown [\underline{5}, 4.6(i); \underline{14}, 2.13.32] that every ∗-primitive associative algebra is a subdirect product of a primitive algebra and its opposite.

0.8 A one-sided primitive (resp. ∗-primitive) associative algebra is necessarily prime (resp. ∗-prime), in particular semiprime [\underline{5}, 4.4].

0.9 Recall the notion of unital hull of an associative algebra R, which is the formal adjoining of an identity to R, i.e., the direct sum Φ1⊕R, where the product is defined to make 1 act as a unit element. We will rather consider the so-called tight unital hulls R̂ of R obtained from the above formal unital hull by factoring out a maximal ideal I of Φ1 ⊕ R not hitting R. Notice that R imbeds into R̂, so that we can assume that R is contained in R̂. Moreover, R is an ideal of R̂ hitting any nonzero ideal of R̂ and R̂ = Φ1 + R.

Similarly ∗-tight unital hulls R̂ can be obtained from any associative algebra R with involution, so that R is a ∗-ideal of R̂ hitting any nonzero ∗-ideal of R̂ and R̂ = Φ1 + R.

Tight and ∗-tight unital hulls have a nice behaviour concerning primitivity and polynomial identities: Let R be an associative algebra (resp. an associative algebra with involution) and let R̂ be a tight (resp. ∗-tight) unital hull of R. Then

(i) R is left (right) primitive (resp. ∗-primitive) if and only if R̂ is left (right) primitive (resp. ∗-primitive). Moreover, if K is a left (right) primitizer (resp. ∗-primitizer) of R with modulus e, then K̂ = Φ(1 − e) + K is a left (right) primitizer (resp. ∗-primitizer) of R̂ (see [\underline{5}, 4.5] which has a straightforward translation for algebras without involution).

(ii) R is PI if and only if R̂ is PI: it is obvious that R̂ being PI implies that R is PI since R ⊆ R̂; the converse follows from the fact that R is a PI ideal of R̂ such that the quotient R̂/R is spanned over Φ by one element, hence is commutative (therefore if f(x_{1}, …, x_{n}) vanishes on R, then f([x_{1}, y_{1}], …, [x_{n}, y_{n}]) vanishes on R̂).

0.10 Primitive Jordan algebras are defined in [\underline{8}] following the corresponding notions of primitivity of associative algebras in terms of modular one-sided ideals (0.5)(iii)(iv). Thus, a Jordan algebra J is primitive if there exits a proper inner ideal K (K ≠ J), called a primitizer of J such that (cf. [\underline{5}, 0.6]):

(i) K is e-modular for some modulus e ∈ J [\underline{8}, 2.2, 2.3, 2.4].

(ii) K complements all nonzero ideals of J (I + K = J for any nonzero ideal I of J).

0.11 Remark: Notice that the notions of primitizers that we have just given correspond to the “old” notions of protoprimitizers given in [\underline{3}, \underline{4}, \underline{5}].

0.12 One can obtain Jordan algebras from associative algebras by symmetrization: If R is an associative algebra, we obtain a Jordan algebra denoted by R^{(+)}, over the same Φ-module, with products built out of the associative product by x^{2} = xx, U_{x}y = xyx, for any x, y ∈ R.

A Jordan algebra is said to be special if it is a subalgebra of R^{(+)} for some associative algebra R.

0.13 Particularly important examples of special Jordan algebras are ample subspaces of associative algebras with involution: If R is an associative algebra with involution ∗, a Φ-submodule H contained in the set of symmetric elements H(R, ∗) is said to be an ample subspace of R if it contains all traces and norms of the elements of R (x + x^{∗}, xx^{∗} ∈ H for any x ∈ R) and xHx^{∗} ⊆ H for any x ∈ R [\underline{9}, p. 387; \underline{11}, 0.8’]. Obviously, H is a subalgebra of R^{(+)}. Ample subspaces of an associative algebra R with involution are usually denoted by H_{0}(R, ∗).

0.14 Given an associative algebra R, the homotope R^{(a)} of R at a ∈ R is the associative algebra over the same Φ-module as R with product x ·_{a} y = xay, for any x, y ∈ R. The subset Ker a = Ker_{R} a = {x ∈ R | axa = 0} is an ideal of R^{(a)} and the quotient R_{a} = R^{(a)}/ Ker a is called the local algebra of R at a.

0.15 Basic results on local algebras of associative and Jordan algebras can be found in [\underline{2}, \underline{6}, \underline{12}]. We stress the following local inheritance of primitivity for associative algebras [\underline{6}, 0.1(iii, v)]: Let R be an associative algebra (resp. an associative algebra with involution ∗), and let 0 ≠ a ∈ R (resp. 0 ≠ a ∈ H(R, ∗)). If R is left (right) primitive (resp. ∗-primitive), then R_{a} is left (right) primitive (resp. ∗-primitive).

1. Associative Algebras

In this section we will obtain the main result of the paper for associative algebras. Apart from its intrinsic interest, this result will play a key role in the proof of the results on Jordan algebras established in the next section.

We begin by establishing the associative version of [\underline{4}, 2.2]. The proof just consists of polishing the proof of that result, eliminating the Jordan algebra considerations. On the other hand, we will need some extra effort to handle the case of an algebra with involution at the same time.

1.1 Recall the well-known result by Kaplansky which establishes that a one-sided primitive PI associative algebra is simple and finite-dimensional over its center, hence unital [\underline{7}, Th. 6.3.1]. Using the description of ∗- primitive algebras in terms of primitive algebras (0.7), one obtains that a PI ∗-primitive associative algebra R is necessarily ∗-simple and unital: Assume first that R is unital. We can use Zorn’s Lemma to find an ideal P of R which is maximal among the ideals L of R satisfying L ∩ L^{∗} = 0; now R/P is one-sided primitive (see the proof of [\underline{5}, 4.6(i)]), hence simple by Kaplansky’s Theorem since it is PI. Let I be any nonzero ∗-ideal of R; clearly I ⊈ P since P ∩ P^{∗} = 0, hence (I + P)/P is a nonzero ideal of R/P, and I + P = R by the simplicity of R/P. Moreover, R = R^{∗} = I^{∗} + P^{∗} = I + P^{∗} and R = R1 ⊆ RR = (I + P)(I + P^{∗}) ⊆ I + P P^{∗}, but P P^{∗} ⊆ P ∩ P^{∗} = 0, and we have shown that I = R. For a not necessarily unital R, take a ∗-tight unital hull R̂ which is also a PI ∗-primitive algebra by (0.9), hence ∗-simple. But R is a nonzero ∗-ideal of R̂, which means R = R̂ and thus R is unital and ∗-simple.

1.2 Lemma. If R is a left (right) primitive (resp. ∗-primitive) associative algebra having a PI primitizer (resp. ∗-primitizer) then Heart R ≠ 0 is a simple algebra (resp. ∗-Heart R ≠ 0 is a ∗-simple algebra).

Proof: This is trivial if R is PI: By Kaplansky’s Theorem (1.1) R is simple unital, hence Heart(R) = R (resp. R is ∗-simple unital, hence ∗-Heart(R) = R), so assume that R is NOT PI.

Let K be a PI left (right) primitizer (resp. ∗-primitizer) of R, and let p(x_{1}, …, x_{n}) be a polynomial identity satisfied by K. For any nonzero ideal (resp. ∗-ideal) I of R, I + K = R. Thus p(R) = p(I + K) ⊆ I + p(K) since I is an ideal of R, and p(R) ⊆ I since p(K) = 0. We have shown that p(R) is contained in every nonzero ideal (resp. ∗-ideal) of R, hence p(R) ⊆ Heart R (resp. p(R) ⊆ ∗-Heart(R)), but p(R) ≠ 0 since R is not PI.

To establish simplicity (resp. ∗-simplicity), let L be a nonzero ideal of Heart R (resp. ∗-ideal of ∗-Heart R). Let S be the ideal of R generated by L (notice that in the case with involution S is a ∗-ideal). By Andrunakievich’s Lemma [\underline{15}, 8.3 Lemma 11], S^{3} is an ideal (resp. ∗-ideal) of R contained in L, hence contained in Heart R (resp. in ∗-Heart R). Moreover, S^{3} ≠ 0 since R is semiprime by (0.8). Thus S^{3} = Heart R (resp. S^{3} = ∗-Heart(R)), which shows that L = Heart R (resp. L = ∗-Heart(R)). Finally, (Heart R)^{2} ≠ 0 (resp. (∗-Heart(R))^{2} ≠ 0) since R is semiprime by (0.8).

1.3 Lemma. If R is a left (right) primitive (resp. ∗-primitive) unital associative algebra having a PI primitizer (resp. ∗-primitizer) then R is simple (resp. ∗-simple).

Proof: We will begin by emphasizing some general facts that will be needed in the proof:

(I) Any local algebra R_{a^{2}} is a homomorphic image of aRa (resp. ∗- homomorphic image if a^{∗} = a), hence if aRa is PI so is R_{a^{2}}: In general φ(x) = axa is an algebra (resp. ∗-algebra, if a^{∗} = a) epimorphism of R^{(a^{2})} onto aRa since φ(x)φ(y) = axa aya = a(xa^{2}y)a = φ(x ·_{a^{2}} y) so R^{(a^{2})}/ Ker(φ) is isomorphic to aRa. Here Ker(φ) = {x ∈ R | axa = 0} = Ker a (cf. (0.14)) ⊆ Ker(a^{2}), so R^{(a^{2})}/ Ker(a^{2}) = R_{a^{2}} is isomorphic to an image of aRa (and this is a ∗-isomorphism if a^{∗} = a).

(II) If R is primitive (resp. ∗-primitive) and aRa is PI for a^{2} ≠ 0 (resp. for a^{∗} = a and a^{2} ≠ 0) then R_{a^{2}} is a simple (resp. ∗-simple) unital algebra. Indeed, if a^{2} ≠ 0 then R_{a^{2}} is primitive (resp. ∗-primitive) by (0.15), and by (I) it is PI, hence is unital simple (resp. ∗-simple) by (1.1).

(III) Prime kernels are core-less: If R is prime, b ≠ 0, then Core(Ker b) = 0 (analogously for ∗-prime and ∗-cores) [If an ideal I is contained in Ker b then (R̂bR̂)I(R̂bR̂) ⊆ R̂(bIb)R̂ = 0 implies I = 0 by primeness].

(IV) If R is unital and b ≡ 1 mod an ideal I, then Ker b ⊆ I: bxb = 0 implies x = 1x1 ≡ bxb = 0 mod I.

(V) If R is prime with simple nonzero heart H^{†} , and some local algebra R_{b} for b ≠ 0 is simple where Ker b ⊆ H (eg. if R is unital and b ≡ 1 mod H,

† If R is prime and contains some simple ideal I ≠ 0 then I must be the heart (analogously for ∗-prime algebras, ∗-simple ideals and ∗-hearts): given an ideal K of R, K ≠ 0 ⟹ K ∩ I ≠ 0 (by primeness) ⟹ K ∩ I = I (by simplicity of I) ⟹ K ⊇ I. by (IV)), then R = H is simple (analogously for ∗-hearts and ∗-simplicity): By corelessness (III) Ker b is strictly contained in H, so H/ Ker b is a nonzero ideal in R_{b}, hence by simplicity it must be all of R_{b} and H = R.

Now we can complete the proof of the lemma.

We can assume that R is not PI since otherwise we can apply (1.1).

We will consider first the case of algebras with involution. Let K be a left ∗-primitizer which is PI. By (0.5)(v), there exists a modulus e for K in ∗-Heart R (which is a nonzero ∗-ideal by (1.2)). By (0.4), ē = e + e^{∗} − e^{∗}e is a ∗-symmetric modulus for K still lying in ∗-Heart R. Renaming, we can assume that e ∈ H(R, ∗) ∩ ∗-Heart(R).

Set a = 1 − e = a^{∗}, b = a^{2}. If b = 0 then 1 ∈ ∗-Heart(R) and ∗-Heart(R) = R as desired. Otherwise 0 ≠ b ≡ 1 mod ∗-Heart(R), where R_{b} is unital ∗-simple by (II) since aRa = (1 − e)R(1 − e) ⊆ R(1 − e) ⊆ K is PI. So R = ∗-Heart(R) by (V). In particular, R is ∗-simple by (1.2).

The case without involution follows formally as above deleting the symmetrization of the modulus and “∗” all over.

1.4 Theorem. Let R be a left (right) primitive (resp. ∗-primitive) associative algebra. Then R has a PI primitizer (resp. ∗-primitizer) if and only if R is simple (resp. ∗-simple) and unital.

Proof: Take a tight (resp. ∗-tight) unital hull R̂ of R. By (0.9)(i), R̂ is left (right) primitive (resp. ∗-primitive) and K̂ = Φ(1−e)+K is a primitizer (resp ∗-primitizer) of R̂ if K is a primitizer (resp. ∗-primitizer) of R with modulus e. Moreover, K is an ideal of K̂: if, for example, K is a left ideal (1 − e)K ⊆ K + eK ⊆ K, and, for any k ∈ K, k(1 − e) = k − ke ∈ K since K is e-modular. Now if K is PI then K̂ is PI just as in (0.9)(ii) (K̂/K is again commutative since spanned by a single element (1 − e) + K). Therefore R̂ is ∗-simple by (1.3). Since R is a nonzero ∗-ideal of R̂, we have R = R̂, hence R is simple and unital.

Conversely, if R is simple (resp. ∗-simple) and unital then it is immediate that R is left and right primitive (resp. ∗-primitive) with primitizer K = 0 (with modulus 1) which is clearly PI.

Once we have (1.4), we proceed to study simple (resp. ∗-simple) unital associative algebras.

1.5 Proposition. If R is a simple unital associative algebra having a nonzero PI one-sided ideal K then R is PI. Moreover, R is finite dimensional over its center.

Proof: We will show directly that R is finite dimensional over its center Z(R), from which being PI is immediate. Replacing R by its opposite if K is a left ideal, we may assume that K is a right ideal.

Since R is simple and unital it is left primitive, hence R is a dense ring of left linear transformations of a vector space over a division algebra by (0.5), i.e., we can assume that R is contained in the ring End_{Δ}(V) of linear transformations (acting on the left) of a right vector space V over a division algebra Δ so that R acts densely on V.

(I) For any 0 ≠ v ∈ V, K(v) = K(V): K(v) ⊇ (KR)(v) = K(Rv) = K(V) since Rv = V by irreducibility of R on V.

(II) K acts densely on W = K(V) over Δ: Given Δ-independent w_{1}, …, w_{n} ∈ W and arbitray u_{1}, …, u_{n} ∈ W, since W = K(w_{1}) by (I), u_{i} = k_{i}(w_{1}), for some k_{i} ∈ K, i = 1, …, n. By density of R on V, there are r_{1}, … r_{n} ∈ R such that r_{i}(w_{j}) = δ_{ij}w_{1}, i, j = 1, …, n, hence k = ∑_{i=1}^{n} k_{i}r_{i} ∈ K satisfies k(w_{i}) = k_{i}r_{i}(w_{i}) = k_{i}(w_{1}) = u_{i}, i = 1, …, n.

(III) The dimension of W over Δ is finite (indeed, at most the PI degree n of K), hence, since W = K(V), all k ∈ K have finite rank: Let

p(x_{1}, …, x_{n}) = x_{1} … x_{n} + ∑_{1≠σ∈S_{n}} α_{σ}x_{σ(1)} … x_{σ(n)}

(α_{i} ∈ Φ) be a multilinear polynomial identity satisfied by K. If the dimension of W is bigger than n, we can find Δ-linearly independent e_{1}, …, e_{n+1} ∈ W, and there exist, using the density of K on W, elements f_{i} ∈ K for i = 1, …, n, such that f_{i}(e_{j}) = δ_{i,j}e_{i+1}, i, j = 1, …, n. It is immediate that (p(f_{n}, …, f_{1}))(e_{1}) = e_{n+1} ≠ 0, hence p(f_{n}, …, f_{1}) ≠ 0, which is a contradiction.

(IV) R = End_{Δ}(V): The set of finite-rank transformations

A = {g ∈ R | dim_{Δ}(g(V)) < ∞}

is clearly an ideal of R, and we have shown that K ⊆ A. In particular A is nonzero. Since R is simple, R = A and the image 1(V) of the unit element 1 of R is finite dimensional. But 1 acts as the identity of V [using the density of R on V, for any v ∈ V, there exists f_{v} ∈ R such that f_{v}(v) = v; hence 1(v) = 1(f_{v}(v)) = (1f_{v})(v) = f_{v}(v) = v], which implies that V is finite dimensional over Δ, and thus R = End_{Δ}(V).

(V) Δ is finite dimensional over its center Z(Δ), which implies by (IV) that R is finite dimensional over its center Z(R) ≅ Z(Δ):

Fix 0 ≠ f_{0} ∈ K. There is some v ∈ V such that f_{0}(v) ≠ 0 and we can find g ∈ R = End_{Δ}(V) such that g(f_{0}(v)) = v. Hence f_{0}(g(f_{0}(v))) = f_{0}(v) and we have found 0 ≠ w = f_{0}(v) and 0 ≠ h = f_{0}g ∈ Kg ⊆ K such that h(w) = w.

Let B = {f ∈ K | f(w) ∈ wΔ}. It is straightforward that B is a Φ-subalgebra of K. Define the map ψ : B ⟶ Δ by ψ(f) = α such that f(w) = wα. It is readily seen that ψ is a Φ-algebra homomorphism and moreover, it is surjective: Let α ∈ Δ and let g ∈ R = End_{Δ}(V) such that g(w) = wα. We have already found h ∈ K such that h(w) = w. Hence hg(w) = h(wα) = wα and hg ∈ Kg ⊆ K since K is a right ideal. Therefore Δ is isomorphic to B/ Ker(ψ), hence Δ is PI since B ⊆ K which is PI.

Notice that Δ is a primitive algebra since it is simple and unital, hence, by Kaplansky’s Theorem (1.1), it is finite dimensional over its center.

1.6 Corollary. If R is a ∗-simple unital associative algebra having a nonzero PI one-sided ideal K then R is PI. Moreover, R is finite dimensional over its ∗-center.

Proof: If R is simple, then R is finite dimensional over its center Z(R) by (1.5). Now, the restriction σ of the involution ∗ to Z(R) is a field automorphism of period two, and H(Z(R), ∗) is just the subfield of invariants of the group generated by σ. Hence the degree of Z(R) over H(Z(R), ∗) is at most two and R is also finite dimensional over H(Z(R), ∗).

If R is not simple it is well-known that R = S ⊕ S^{∗} ≅ S ⊕ S^{op} for simple S, and S is unital since R is so.

Let K be a nonzero PI one-sided ideal of R. By projecting on each direct summand of R ≅ S ⊕ S^{op} we obtain a nonzero PI one-sided ideal either in S or in S^{op}. Therefore, by (1.5), either S or S^{op} is PI, hence both S and S^{op} are PI. Moreover, S is finite dimensional over its center Z(S) and S^{op} is finite dimensional over its center Z(S^{op}) ≅ Z(S). Notice that Z(R) ≅ Z(S)⊕Z(S) and R is finite dimensional over the field H(Z(R), ∗) = {t ⊕ t^{∗} | t ∈ Z(S)} ≅ Z(S).

Putting together (1.4), (1.5) and (1.6) yields the main result in this paper concerning associative algebras:

1.7 Theorem. If R is a left (right) primitive (resp. ∗-primitive) associative algebra having a nonzero PI primitizer (resp. ∗-primitizer) then R is PI. Moreover R is finite dimensional over its center (resp. over its ∗-center).

2. Jordan Algebras

In [\underline{4}, 2.2] it is shown that a special primitive Jordan algebra having a PI primitizer is necessarily simple and unital. Since the only exceptional primitive algebras are Albert algebras [\underline{5}, 5.1], which are PI, simple and unital, we can remove speciality in the previous assertion. Moreover, its converse has the same proof as the corresponding fact for associative algebras (1.4). So, we readily obtain a Jordan version of (1.4).

2.1 Theorem. Let J be a primitive Jordan algebra. Then J has a PI primitizer if and only if J is simple and unital.

2.2 The link between (1.5), (1.6) and their analogue for Jordan algebras is given by Amitsur in [\underline{1}, Th. 1] from which the following fact is obtained: If an associative algebra with involution satisfies a polynomial identity of the form p(x_{1}, …, x_{n}; x_{1}^{∗}, …, x_{n}^{∗}), then it is PI.

2.3 Proposition. If J is a simple unital Jordan algebra having a PI inner ideal K with K^{2} ≠ 0 then J is PI.

Proof: In the classification of simple Jordan algebras [\underline{11}, 15.5], the only ones not necessarily PI are those of the form H_{0}(R, ∗), where R is a ∗-simple associative algebra. We will then assume that J has such form.

We first remark that R is unital with the same unit element as J (see the proof of [\underline{5}, 4.9]).

Let K be a PI inner ideal of J with K^{2} ≠ 0 and fix k ∈ K such that k^{2} ≠ 0. We will show that the nonzero left ideal Rk of R satisfies a polynomial identity involving the involution as in (2.2). An element c ∈ Rk has the form c = rk, for r ∈ R, hence, using that k ∈ K ⊆ H_{0}(R, ∗):

c^{∗}c = kr^{∗}rk = U_{k}(r^{∗}r) ∈ U_{k}(H_{0}(R, ∗)) = U_{k}J ⊆ U_{K}J ⊆ K

since K is an inner ideal of J. Now, if K satisfies the polynomial identity p(x_{1}, …, x_{n}) then Rk satisfies p(x_{1}^{∗}x_{1}, …, x_{n}^{∗}x_{n}). In particular kRk satisfies p(x_{1}^{∗}x_{1}, …, x_{n}^{∗}x_{n}), hence kRk is PI by (2.2). We remark that (2.2) cannot be applied directly to Rk since the involution can not be restricted to Rk in general.

Let q(x_{1}, …, x_{m}) be a polynomial identity satisfied by kRk. For any r_{1}, …, r_{m} ∈ R, q(kr_{1}k, …, kr_{m}k) = 0. Thus, for any r_{m+1} ∈ R,

kq(kr_{1}k, …, kr_{m}k)kr_{m+1} = 0.

Notice that

kq(kr_{1}k, …, kr_{m}k)kr_{m+1} = q(k^{2}r_{1}, …, k^{2}r_{m})k^{2}r_{m+1},

hence we have shown that the nonzero right ideal k^{2}R satisfies the polynomial identity q(x_{1}, …, x_{m})x_{m+1}. By (1.6), R is finite dimensional over a field F contained in the center of R.

We just need to show that the above implies that J is PI. Indeed, if dim_{F} (End_{F} R) = (dim_{F} R)^{2} = m − 1, then for any a_{1}, …, a_{m}, b ∈ R, the operators U_{a_{1},b}, …, U_{a_{m},b} (defined on R) are F -linearly dependent elements of End_{F} R (they are F -linear since they are built out of one-sided associative multiplication operators and F is contained in the center of R), hence s_{m}(U_{a_{1},b}, …, U_{a_{m},b}) = 0, where s_{m} is the standard polynomial identity of degree m [\underline{7}, Section 2 of Chapter 6], and R and, in particular, J satisfy the Jordan polynomial identity

f(x_{1}, …, x_{m}; y, z) = s_{m}(U_{x_{1},y}, …, U_{x_{m},y})(z).

2.4 The final step in the proof of (2.3) also follows from this fact: If a special Jordan algebra imbeds in an associative PI algebra satisfying f(x_{1}, …, x_{n}) then it is a Jordan PI algebra since it satisfies

g(x, y) = f(xyx, xy^{2}x, … xy^{n}x)f(xyx, xy^{2}x, … xy^{n}x)^{reverse}.

This latter is a Jordan polynomial by Cohn’s Theorem [\underline{10}, page 763], since it is symmetric and involves only two variables.

From (2.1) and (2.3), our central result concerning Jordan algebras is obtained.

2.5 Theorem. If J is a primitive Jordan algebra having a PI primitizer K with K^{2} ≠ 0 then J is PI.

3. Final Remarks

3.1 The following example shows that the condition of K^{2} ≠ 0 in (2.3) and (2.5) cannot be deleted. Take Δ an associative division algebra of infinite dimension over its center, and let Δ_{n} be the simple unital (hence primitive) associative algebra of n × n matrices over Δ (n > 1). Its symmetrization (Δ_{n})^{(+)} is a simple [\underline{9}, Th. 4] unital Jordan algebra (hence primitive). Let us consider the nonzero inner ideal

K = e_{12}Δ_{n}e_{12} = e_{12}Δ.

It is clear that K is a nonzero primitizer of (Δ_{n})^{(+)}. Moreover, K is PI since K^{2} = 0. However, (Δ_{n})^{(+)} is not PI since otherwise Δ_{n}, hence Δ, would be PI and then finite dimensional over its center by (1.1).

3.2 Kaplansky’s Theorem [\underline{7}, Th. 6.3.1] gives a bound on the dimension of a PI primitive associative algebra over its center in terms of the degree of the identity satisfied by the algebra. This is not the case when dealing with the polynomial identities satisfied by a nonzero primitizer. As an example, let us consider R = F_{n}, the associative F -algebra of n × n matrices over a field F. Here Z(R) ≅ F and dim_{Z(R)} R = n^{2}. Let K = e_{11}R be the span F (e_{11}, …, e_{1n}) of the matrix units e_{11}, … e_{1n}. It is clear that K is a nonzero right ideal of R. Moreover, K is 1-modular and, since R is simple, it is a primitizer of R. One can readily check that if K = eR for any reduced idempotent (eRe = F e) then K satisfies the degree three polynomial identity

p(x, y) = xyy − yxy,

(because [x, y]y = [er, es]es = [ere, ese]es = [αe, βe]es = 0) while dim_{Z(R)} R can be as big as desired.

3.3 Polynomial identities in one-sided ideals (resp. inner ideals) of an associative (resp. Jordan) algebra are clearly related to polynomial identities satisfied by homotopes of the algebra, hence polynomial identities in local algebras: if, for example, an inner ideal K of a Jordan algebra J satisfies the polynomial identity p(x_{1}, …, x_{n}), then, p(U_{k}a_{1}, …, U_{k}a_{n}) = 0, for any k ∈ K and a_{1}, …, a_{n} ∈ J, which implies that the homotope J^{(k^{2})} satisfies the polynomial identity U_{x_{n+1}}p(x_{1}, …, x_{n}).

In his recent paper [\underline{13}], F. Montaner shows that if a simple unital Jordan algebra J has a PI local algebra then it satisfies a homotope polynomial identity which depends on the capacity of J [\underline{13}, 2.6(i)]; in particular, J is PI. This provides a different proof of (2.5) in which purely Jordan techniques are used. Montaner’s work involves a deeper use of the classification of simple Jordan algebras [\underline{11}, 15.5] connected to the notion of capacity, instead of our approach based on the translation of results between associative and Jordan algebras.

ACKNOWLEDGEMENTS

The authors wish to thank professors Teresa Cortés and Fernando Montaner for their very valuable comments during the preparation of this paper. They also thank the referee for his or her careful reading of the manuscript, correcting some mistakes and suggesting very interesting improvements.

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