On Polynomial Identities in Associative and Jordan Pairs

Algebr Represent Theor (2010) 13:189–205 DOI 10.1007/s10468-008-9114-5 On Polynomial Identities in Associative and Jordan Pairs Fernando Montaner · Irene Paniello Received: 12 May 2007 / Accepted: 22 February 2008 / Published online: 12 February 2009 © Springer Science + Business Media B.V. 2009 Abstract We prove that a Jordan system satisfies a polynomial identity if and only if it satisfies a homotope polynomial identity. In the obtention of that result, we also prove an analogue for associative pairs with involution of Amitsur’s theorem on associative algebras satisfying a polynomial identity with involution. Keywords Polynomial identity · Jordan pairs · Involution Mathematics Subject Classifications (2000) 17C05 · 17C10 · 16R50 · 16R20 1 Introduction Polynomial identities play a basic structural role in nonassociative theory. In contrast with the study of associative rings satisfying polynomial identities, where that Presented by Susan Montgomery. Fernando Montaner was partially supported by the Spanish Ministerio de Ciencia y Tecnología and FEDER (MTM 2004-08115-CO4-02), and by the Diputación General de Aragón (Grupo de Investigación de Álgebra). Irene Paniello was partially supported by the Spanish Ministerio de Ciencia y Tecnología and FEDER (MTM 2004-08115-CO4-02). F. Montaner (B) Departamento de Matemáticas, Universidad de Zaragoza, 50.009 Zaragoza, Spain e-mail: [email protected] I. Paniello Departamento de Estadística e Investigación Operativa, Universidad Pública de Navarra, 31006 Pamplona, Spain e-mail: [email protected] 190 F. Montaner, I. Paniello condition can be understood as a kind of finiteness condition which allows a strengthened form of the structural results of the general theory, its nonassociative counterpart is an unavoidable ingredient in the obtention of general structure theories. In particular, that is the case in Zelmanov’s fundamental results [18] on Jordan algebras, where the existence of the so-called hermitian polynomials leads to the study of algebras that satisfy a particular kind of identities (Clifford identities) prior to the obtention of the general classification theorem. For Jordan pairs and triple systems the situation is entirely analogous to the algebra case. The only difference is that the role played by polynomial identities is now played by homotope polynomial identities (see [6–8, 20, 21]), that is, polynomial identities that hold in all homotope algebras of the system. This partially motivated the study of “local PItheory” in [20–22], that is, the study of Jordan systems with local algebras satisfying a polynomial identity, and in particular, of Jordan systems satisfying a homotope polynomial identity. In spite of the effectiveness of the use of homotope polynomials in the structural results, usual identities deserve some attention, both because they are conceptually simpler, and because they seem easier to obtain than homotope identities. In that respect, the study of graded polynomial identities on 3-graded Lie algebras, which will be the subject of a forthcoming paper by the authors, naturally leads through the Kantor–Koecher–Tits construction, to general identities in Jordan pairs. On the other hand, the result of Zelmanov [26, Theorem 3] asserting that in a PI-Jordan system, the McCrimmon radical and the nil-radical coincide, suggests that some significant results could be reached for general polynomial identities. The question of whether a theory of general identities of Jordan systems could be developed remained however open, and was the content of a question raised in [21], namely: does every PI Jordan system satisfy a homotope polynomial identity? In this paper we answer in the affirmative that conjecture. Since our approach makes use of the structure theory, we need to gain first some information on Jordan pairs of hermitian elements H(A, ∗) for an associative pair A with involution ∗. Thus, after a section of preliminaries, we devote a section to the study of associative pairs satisfying a ∗-polynomial identity (a polynomial identity with involution). We prove there a pair analogue of Amitsur’s celebrated result [1] on algebras with involution satisfying a ∗-polynomial identity. Our proof is closely patterned after the proof of the algebra result as exposed by Herstein in [12], which is due to Montgomery (see [12, page 185]). Apart from its own interest, that result is instrumental in the final section for the proof of the affirmative answer to the foregoing question. Finally, we combine that positive answer with the local PI-theory to obtain Jordan analogues of Kaplansky’s and Posner-Rowen’s theorems. In view of the approach followed in the paper, one may ask whether the structural results on PI-Jordan systems could be reached directly without appealing to the local PI-theory. This might well be so, but a careful analysis of the associative GPItheory uncovers the role played in it by local algebras (see for instance [24] or [25, Chapter 7], and specially the approach followed in [3]). On the other hand, looking at the standard embedding of associative systems, the PI-condition seems to be intermediate between a PI and a GPI condition, and this suggests that, one way or another, local PI-algebras should make its appearance in their study. On polynomial identities in associative and Jordan pairs 191 2 Preliminaries 2.1 We will work with associative and Jordan systems over a unital commutative ring of scalars  which will be fixed throughout. We refer to [13, 18, 19] for notation, terminology and basic results. We recall in this section some of those notations and basic results. 2.2 An associative pair over  is a pair A = (A+ , A− ) of -modules together with trilinear maps  , , σ : Aσ × A−σ × Aσ → Aσ (x, y, z)  → x, y, zσ such that x, y, zσ , u, vσ = x, y, z, u−σ , vσ = x, y, z, u, vσ σ for all x, z, v ∈ Aσ , y, u ∈ A−σ and σ = ±. An involution in the associative pair A is a pair of linear maps ∗ : Aσ → Aσ x  → x∗ such that (x∗ )∗ = x and x, y, z∗ = z∗ , y∗ , x∗  for all x, z ∈ Aσ , y ∈ A−σ , σ = ±. To make the notation less cumbersome, we will usually denote the products in associative pairs simply by juxtaposition, so that if a, c ∈ Aσ , and b ∈ A−σ for σ = ±, abc will mean a, b , cσ . We recall that associative pairs have a standard imbedding into an associative algebra where the juxtaposition of any two factors makes sense. Let A be an associative pair. An A-module is a pair of -modules (M+ , M− ) endowed with two bilinear maps M σ × Aσ → M−σ (m, x)  → mx for σ = ±, that satisfy ((mx)y)z = m(xyz) for all x, z ∈ Aσ , y ∈ A−σ , and m ∈ Mσ . It is clear how to define the notion of A-submodule of an A-module, and the notions of irreducible and faithful A-modules. 2.3 Given any associative pair A = (A+ , A− ) we denote by U A , the standard imbedding of A. Recall that U A is a unital associative algebra with two idempotents e1 + e2 = 1 such that if we consider the Peirce decomposition of U A with respect to these idempotents, then A = ((U A )12 , (U A )21 ) with the usual triple product. Every involution of A extends uniquely to an algebra involution of U A that satisfies e∗1 = e2 [9, 3.2]. 192 F. Montaner, I. Paniello 2.4 The socle of A = (A+ , A− ) is Soc(A) = (Soc(A+ ), Soc(A− )), where Soc(Aσ ) is the sum of all minimal right ideals of Aσ . If A has no minimal right ideals, we write Soc(A) = 0. An associative pair A has finite capacity if it satisfies both the ascending and the descending chain condition on principal inner ideals (see [13] for definitions). In that case A equals its socle and it contains a maximal idempotent: an idempotent e = (e+ , e− ) whose Peirce 00-space A00 vanishes (see [13]). 2.5 Any element a ∈ A−σ , determines a homotope algebra (Aσ )(a) , an associative algebra over the -module Aσ with multiplication x ·a y = xay for any x, y ∈ Aσ . The set Ker A a = Ker a = {x ∈ Aσ | axa = 0} is an ideal of (Aσ )(a) , and the quotient Aaσ = (Aσ )(a) /Ker a is an associative algebra called the local algebra of A at a. 2.6 Let X = (X + , X − ) be a pair of nonempty sets. We denote by FAP(X) the free associative pair over  on X, and call its elements (pair) polynomials. Note that FAP(X) is the subpair of the pair (FA(X + ∪ X − ), FA(X + ∪ X − )) obtained by doubling the free associative algebra FA(X + ∪ X − ), generated by (X + , X − ). Clearly, every nonzero pair polynomial has odd degree. By the universal property of FAP(X), − + − any polynomial f (x+ 1 , . . . , xn , x1 , . . . , xn ) can be evaluated in an associative pair σ σ A on fixed values xi = ai ∈ Aσ for the indeterminates xσ ∈ X σ . An associative polynomial fσ ∈ FAP(X)σ is a polynomial identity of an associative pair A, and A is then said to be a PI-associative pair, if fσ is monic in the sense that some leading monomial in fσ has coefficient 1, and all the evaluations of fσ in A vanish. A polynomial pσ ∈ FAP(X)σ of degree m = 2d + 1, and involving the variables −σ xσ1 , . . . , xσd+1 and x−σ 1 , . . . , xd is multilinear if each monomial appearing in pσ (note σ that FA(X) is a free -module over the set of monomials) its degree in each variable x±σ ∈ X is exactly 1. As for algebras, it is easy to see that if an associative pair A satisfies a polynomial identity of degree m, then it satisfies a multilinear identity of degree m. 2.7 Let X = (X + , X − ) and Z = (Z + , Z − ) be two pairs of sets and assume that there are bijections ∗ : X σ → Z σ (whose inverse we also denote by ∗). Then the associative pair FAP(X, Z ) = FAP(X + ∪ Z + , X − ∪ Z − ) can be endowed with an involution extending ∗ in the obvious way. Its elements are called ∗-polynomials. We have a notion of degree for any ∗-polynomial pσ (x+ , x− , (x+ )∗ , (x− )∗ ) = + ∗ − − ∗ + + ∗ − − ∗ σ pσ (x+ 1 , . . . , xn , (x1 ) , . . . , (xn ) , x1 , . . . , xn , (x1 ) , . . . , (xn ) ) ∈ FAP(X, Z ) = FAP ∗ σ + − (X, X ) . An associative pair A = (A , A ) with involution ∗ satisfies pσ (x+ , x− , (x+ )∗ , (x− )∗ ) as before if that polynomial vanishes under every substitution xiσ ∈ Aσ (see [12, p. 185]). An associative pair is a ∗-PI-associative pair if it + ∗ + + ∗ satisfies a monic ∗-polynomial. A ∗-polynomial pσ (x+ 1 , . . . , xn , (x1 ) , . . . , (xn ) , On polynomial identities in associative and Jordan pairs 193 − ∗ − − ∗ ∗ x− 1 , . . . , xn , (x1 ) , . . . , (xn ) ) ∈ FAP(X, X ) is multilinear if the polynomial − − + + − + + pσ (x1 , . . . , xn , z1 , . . . , zn , x1 , . . . , xn , z1 , . . . , z− n ) ∈ FAP(X ∪ Z ) is multilinear. 2.8 Lemma Let A = (A+ , A− ) be an associative pair endowed with an involution ∗, and satisfying a ∗-polynomial identity pσ (x+ , x− , (x+ )∗ , (x− )∗ ) = mσ (x+ , x− , (x+ )∗ , (x− )∗ ) + · · · , of degree 2d + 1, where mσ (x+ , x− , (x+ )∗ , (x− )∗ ) is a monomial of degree 2d + 1. Then A satisfies  pσ (x+ , x− , (x+ )∗ , (x− )∗ ) = xσ1 x−σ 1 ... σ −σ σ + − + ∗ − ∗ xd xd xd+1 + qσ (x , x , (x ) , (x ) ), where each monomial of qσ (x+ , x− , (x+ )∗ , (x− )∗ ) is of degree 2d + 1, involves each xiσ or (xiσ )∗ for every σ ∈ {+, −}, but not σ + − + ∗ − ∗ both, and where xσ1 x−σ 1 . . . xd+1 does not occur in qσ (x , x , (x ) , (x ) ). Therefore, A satisfies a multilinear identity of degree 2d + 1. Proof See [12, Lemma 5.1.1]. ⊓ ⊔ 2.9 A Jordan algebra has products U x y and x2 , quadratic in x and linear in y, whose linearizations are U x,z y = Vx,y z = {x, y, z} = U x+z y − U x y − U z y and x ◦ y = Vx y = (x + y)2 − x2 − y2 . A Jordan pair V = (V + , V − ) has products Qx y for x ∈ V σ and y ∈ V −σ , σ = ±, with linearizations Qx,z y = Dx,y z = {x, y, z} = Qx+z y − Qx y − Qz y. A Jordan triple system T has product Px y, whose linearizations are Px,z y = Lx,y z = {x, y, z} = Px+z y − Px y − Pz y. Any Jordan pair V = (V + , V − ) gives rise to a polarized Jordan triple system T(V) = V + ⊕ V − with product Px+ ⊕x− y+ ⊕ y− = Px+ y− ⊕ Px− y+ . Conversely doubling a Jordan triple system T produces a double Jordan pair V(T) = (T, T) with Qx y = Px y for any x, y ∈ T. We denote by Ŵ(J) the centroid of a Jordan system J (see [13, 17] for definitions). If J is a strongly prime Jordan system, then Ŵ(J) is a domain acting faithfully on J, and we can form the central closure Ŵ(J)−1 J as the quotient module (or pair of modules if J is a Jordan pair) of J, which is a Jordan system of the same type as J over the field of fractions Ŵ(J)−1 Ŵ(J) of Ŵ(J). We refer to [21] for the related notion of extended centroid of a Jordan system J, which we denote C (J), and the attached scalar extension (for a nondegenerate J): its extended central closure C (J)J. 2.10 Jordan systems can be obtained from associative systems by symmetrization. Every associative algebra A gives rise to a Jordan algebra A(+) , by taking U x y = xyx and x2 = xx for x, y ∈ A. Similarly, every associative pair A = (A+ , A− ) produces a Jordan pair A(+) simply by defining Qx y = xyx for xσ ∈ Aσ , y−σ ∈ A−σ , σ = ±. A Jordan system (algebra, pair or triple system) is special if it is isomorphic to a Jordan subsystem of A(+) for some associative system A. 194 F. Montaner, I. Paniello 2.11 Associative systems with involution give rise to important examples of special Jordan systems. Given any associative algebra A with involution ∗, the set H(A, ∗) = {a ∈ A | a∗ = a} ⊂ A(+) of symmetric elements of A is a hermitian Jordan algebra. More generally, we can consider ample hermitian subspaces H0 (A, ∗) ⊆ H(A, ∗) of symmetric elements containing all traces {a} = a + a∗ and norms aa∗ of the elements of A and such that aH0 (A, ∗)a∗ ⊂ H0 (A, ∗) for all a ∈ A. Recall that if 12 ∈ , the only ample subspace is H0 (A, ∗) = H(A, ∗) [18]. = (A+ , A− ) is  If A  an associative pair with an involution ∗, then H(A, ∗) = + H(A , ∗), H(A− , ∗) ⊂ A(+) where H(Aσ , ∗) = {a ∈ Aσ | a∗ = a} is a hermitian Jordan pair. An ample hermitian subpair is a subpair H0 (A, ∗) = H0 (A+ , ∗), H0 (A− , ∗) ⊆ H(A, ∗) that contains all traces {a} = a + a∗ of elements a ∈ Aσ and satisfies aH0 (A−σ , ∗)a∗ ⊆ H0 (Aσ , ∗) for all a ∈ Aσ , σ = ±. 2.12 Given a Jordan pair V = (V + , V − ) and a ∈ V −σ the -module V σ becomes a Jordan algebra denoted (V σ )(a) and called the a-homotope of V by defining U x(a) y = Qx Qa y and x(2,a) = Qx a, for any x, y ∈ V σ . The set KerV a = Ker a = {x ∈ V σ | Qa x = Qa Qx a = 0} is an ideal of (V σ )(a) and the quotient Vaσ = (V σ )(a) /Ker a is a Jordan algebra, called the local algebra of V at a. If V is nondegenerate, then Ker a = {x ∈ V σ | Qa x = 0}. 2.13   The socle Soc(V) = Soc(V + ), Soc(V − ) of a nondegenerate Jordan pair V = (V + , V − ) is the sum of all minimal inner ideals of V. The socle is a direct sum of simple ideals [14], therefore Soc(V) is simple if the Jordan pair is strongly prime. As for associative pairs, a Jordan pair V has finite capacity if it satisfies both the ascending and the descending chain condition on inner ideals. Again in this case, V equals its socle, and contains a maximal idempotent e, an idempotent whose Peirce 0-space V0 (e) vanishes (see [15]). 2.14 We refer to [20, 21, 24, 25] for the basic notions on PI-theory for associative and Jordan systems. A polynomial f = f (x1 , . . . , xn ) ∈ FJ(X), the free Jordan algebra on a set X, is called essential if its image in the free special Jordan algebra FSJ(X) under the natural homomorphism has the same degree as f , and has a monic leading term as an associative polynomial (note that FSJ(X) is isomorphically embedded in the symmetrized Jordan algebra FA(X)(+) of the free associative algebra FA(X)) . A Jordan PI-algebra is a Jordan algebra which satisfies some essential f (x1 , . . . , xn ). This definition extends to Jordan pairs by considering the free Jordan pair FJP(X + , X − ) on the sets of generators (X + , X − ) (see [23]). Here one considers the free special Jordan pair FSJP(X + , X − ), which embeds isomorphically into the Jordan pair FAP(X + , X − ), and the natural homomorphism τ : FJP(X + , X − ) → On polynomial identities in associative and Jordan pairs 195 FSJP(X + , X − ) extending the identity on X σ , σ = ±, and defines an essential polynomial as a nonzero polynomial f ∈ FJP(X + , X − ) such that τ ( f ) has the same degree as f , and has a monic coefficient as an element of FAP(X + , X − ). A PI-Jordan pair is then a Jordan pair satisfying some essential polynomial. This definition extends in the obvious way to Jordan triple systems. If f (x1 , . . . , xn ) 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 FJT(X), the polynomial f (z; x1 , . . . , xn ) = f (z) (x1 , . . . , xn ) is the image of f under the only homomorphism FJ(X) → FJT(X)(z) extending the identity on X [6, 7]. A Jordan triple system T satisfies a homotope polynomial identity (homotope-PI or HPI, for short) if there exists an essential polynomial f (x1 , . . . , xn ) in FJ[X] such that f (y; x1 , . . . , xn ) with y ∈ X different from xi vanishes under all substitutions of elements y, xi ∈ T. This definition extends to Jordan pairs V by considering polarized triple systems. + − Indeed, since for every a+ ⊕ a− ∈ T(V) the homotope T(V)(a ⊕a ) is isomorphic to + (a− ) − (a+ ) the product (V ) × (V ) , a polynomial f (x1 , . . . , xn ) ∈ FJ[X] is an identity of all homotopes of T(V) if and only if it is an identity of all homotopes of V. Note that a homotope polynomial identity on a Jordan pair is a pair of usual identities: If V satisfies f (y; x1 , . . . , xn ) for an admissible Jordan polynomial f , then it satisfies g−σ = f (y−σ ; xσ1 , . . . , xσn ) for σ = ± (see however [10], where it is proved that if V satisfies some f (y−σ ; xσ1 , . . . , xσn ) = 0 for a fixed σ = + or −, then it satisfies a homotope polynomial identity). 2.15 The fact that a Jordan system J satisfies a homotope-PI means that all homotopes, and consequently all local algebras, satisfy a given identity. Nonetheless, sometimes we will also be interested in the existence of some aσ ∈ V σ such that the local algebra Va−σ is PI. Such elements are called PI-elements. If V is a Jordan pair we write σ PI(V) = (PI(V + ), PI(V − )) for the set of PI-elements of V. It was proved in [20, 5.4] that PI(J) is an ideal in every nondegenerate Jordan system J. 3 Polynomial Identities in Associative Pairs The purpose of this section is to relate the existence of polynomial and homotope polynomial identities on associative pairs endowed with involution. From now on we will assume those polynomial identities are as in 2.8. We begin with primitive pairs. 3.1 Let A = (A+ , A− ) be a primitive associative pair. By the Density Theorem [5, Theorem 1], there is a division -algebra  and two nonzero -vector spaces M+ , M− such that A is isomorphic to a dense subpair of H = (Hom (M− , M+ ), Hom (M+ , M− )). Besides the standard imbedding U A of A is a primitive associative algebra and M = M− ⊕ M+ is a faithful irreducible right U A -module such that  is isomorphic to End (MU A ). 196 F. Montaner, I. Paniello 3.2 Lemma Let A = (A+ , A− ) be an associative pair, and let (M+ , M− ) be a faithful irreducible A-module. Then either A has a minimal right ideal or given two finite dimensional vector spaces W σ ⊆ Mσ , σ = ±, and v σ ∈ Mσ a vector not contained in W σ , there exists a−σ ∈ A−σ such that W σ a−σ = 0 and v σ a−σ  ∈ W −σ . Proof Suppose that A does not have minimal right ideals. Then A has no nonzero elements of finite rank, and therefore, Mσ a−σ is an infinite dimensional vector space for all nonzero a−σ ∈ A−σ . Since W −σ is finite dimensional, this implies that there ⊓ ⊔ exists a−σ ∈ A−σ such that W σ a−σ = 0 and v σ a−σ  ∈ W −σ by [5, Theorem 1]. 3.3 Lemma Let A = (A+ , A− ) be a primitive associative pair with a polarized involution ∗ and (M+ , M− ) be a faithful irreducible A-module. Then (i) either A has minimal right ideals, (ii) or given any two finite dimensional vector subspaces W σ ⊂ Mσ , σ = ±, and any vector v −σ ∈ M−σ not contained in W −σ , there exists an element rσ ∈ Aσ that satisfies the following conditions: (a) W −σ rσ = 0, (b) W −σ (rσ )∗ = 0, (c) v −σ (rσ )∗ = 0, (d) v −σ rσ  ∈ W σ . Proof Suppose that A has no minimal right ideals. Then, by [5, Theorem 2], A has no nonzero elements of finite rank and M−σ xσ is an infinite dimensional -subspace of Mσ for all 0  = xσ ∈ Aσ . Fix σ ∈ {+, −}. Then, given a finite dimensional -space W −σ ⊂ M−σ and v −σ ∈ −σ M not in W −σ , by [5, Theorem 2], there exists a nonzero element aσ ∈ Aσ such that W −σ aσ = 0 and v −σ aσ = 0. Moreover, since W σ is finite dimensional over  and 0  = M−σ (aσ )∗ is an infinite dimensional vector -subspace of Mσ , we can take 0  = uσ ∈ M−σ (aσ )∗ such that uσ  ∈ W σ . Consider now Bσ = {yσ ∈ Aσ | W −σ yσ = 0}, which is a right ideal of A, and satisfies v −σ Bσ  = 0 by 3.2. It follows from the equalities v −σ Bσ = Mσ and ((v −σ Bσ )A−σ )Aσ ⊆ v −σ (Bσ A−σ Aσ ) ⊆ v −σ Bσ , that M = (v −σ Bσ , (v −σ Bσ )A−σ ). Hence there exists b σ ∈ Bσ such that 0  = v −σ b σ and therefore we have M−σ = (v −σ b σ )A−σ . Finally since uσ ∈ M−σ (aσ )∗ , we can write uσ = ((v −σ b σ )x−σ )(aσ )∗ for some x−σ ∈ A−σ and it is easily seen that rσ = b σ x−σ (aσ )∗ satisfies the required properties. ⊓ ⊔ 3.4 Proposition Let A = (A+ , A− ) be a primitive associative pair. a) If A is PI, then A has nonzero socle. b) If A is endowed with an involution ∗, and A is ∗-PI, then A has nonzero socle. On polynomial identities in associative and Jordan pairs 197 Proof Let us first prove b). Suppose that A has no minimal right ideals. Then, with the notation of 3.1, and applying 3.3 to W ± = 0 and an arbitrary nonzero vector v −σ ∈ M−σ , we obtain aσ1 ∈ Aσ such that W −σ aσ1 = 0, v −σ (aσ1 )∗ = 0, W −σ (aσ1 )∗ = 0, v −σ aσ1  ∈ W σ . Since v −σ aσ1  = 0, we can apply 3.3 again with W −σ = v −σ , W σ = 0, and v −σ aσ1  = 0, −σ to obtain an element a−σ such that 1 ∈ A W σ a−σ 1 = 0, ∗ (v −σ aσ1 )(a−σ 1 ) = 0, ∗ W σ (a−σ 1 ) = 0, −σ = v −σ . (v −σ aσ1 )a−σ 1 ∈ W + + We claim that repeated application of 3.3 produces sequences a+ 1 , · · · , am ∈ A − − − and a1 , . . . , am ∈ A , for any m ∈ N, that satisfy the following conditions: σ −σ σ (1.a) ((((. . . (v −σ aσ1 )a−σ 1 ) . . .)ai )ai )a j = 0 for all i + 1 < j ≤ m, σ −σ −σ σ −σ (1.b) (((. . . (v a1 )a1 ) . . .)ai )a j = 0 for all i < j ≤ m, σ −σ σ ∗ (2) (2.a) ((((. . . (v −σ aσ1 )a−σ 1 ) . . .)ai )ai )(a j ) = 0 for all i + 1 ≤ j ≤ m, −σ ∗ σ (2.b) (((. . . (v −σ aσ1 )a−σ 1 ) . . .)ai )(a j ) = 0 for all i ≤ j ≤ m, (3) The sets (1) −σ σ −σ −σ (3.a) {v −σ , (v −σ aσ1 )a−σ a1 )a1 ) . . .)aσm )a−σ , m }⊆ M 1 , . . . , (((. . . (v −σ −σ σ −σ σ −σ σ −σ σ σ (3.b) {v a1 , ((v a1 )a1 )a2 , . . . , ((. . . (v a1 ) . . .)am−1 )am } ⊆ Mσ . are linearly independent over  −σ σ Indeed, take aσ1 , a−σ satisfying (1)–(3). Then, since both U −σ = 1 , . . . , ak , ak −σ −σ σ −σ −σ σ −σ v + ((v a1 )a1 ) + · · · + (((v a1 ) . . .)a−σ and U σ = (v −σ aσ1 ) + k−1 ) ⊆ M σ σ · · · + ((((v −σ aσ1 ) . . .)a−σ )a ) ⊆ M are finite dimensional over  and vk−σ = k−1 k σ σ −σ σ −σ −σ σ (((v a1 ) . . .)ak )ak  ∈ U by 3.3, there exists ak+1 ∈ A such that U −σ aσk+1 = 0, vk−σ (aσk+1 )∗ = 0, U −σ (aσk+1 )∗ = 0, vk−σ aσk+1  ∈ U σ . −σ Continuing in this way, with U σ = U σ , U −σ = U −σ + (((v −σ aσ1 ) . . .)a−σ + k )=U −σ −σ σ −σ σ −σ −σ σ −σ σ −σ vk , and vk+1 = vk ak+1 = ((((vk a1 )a1 ) . . .)ak )ak+1 , we obtain ak+1 ∈ A such −σ −σ σ σ that aσ1 , a−σ 1 ,. . .,ak , ak , ak+1 , ak+1 satisfies (1)–(3), thus proving the claim. We can assume that A satisfies a multilinear polynomial identity pσ of degree 2d + 1 of the form 2.8. We claim that the above inductive process gives rise to a sequence of elements in A on which that polynomial identity does not vanish. Indeed, if we build the above sequence for m = d + 1, and we evaluate pσ in xσ1 = aσ1 , x−σ 1 = −σ −σ σ σ a−σ , . . . , x = a , x = a , we obtain 1 d d d+1 d+1 −σ −σ ∗ σ ∗ 0 = v −σ pσ (aσ1 , . . . , aσd+1 , a−σ 1 , . . . , ad , (a1 ) , . . . , (ad ) ) = σ −σ qσ (aσ1 , . . . , a−σ = (((v −σ aσ1 )a−σ 1 ) . . .)ad+1 + v 1 , . . .). So it suffices to check that v −σ qσ (aσ1 , . . . , a−σ 1 , . . .) = 0, since this would imply σ (((v −σ aσ1 )a−σ ) . . .)a = 0, contrary to (1)–(3). 1 d+1 198 F. Montaner, I. Paniello To do this, first note that any monomial in qσ (aσ1 , . . . , a−σ 1 , . . .) which does not begin with aσ1 annihilates v −σ . Indeed, by (1.a), v −σ aiσ = 0 for all i > 1 and, by (2.a), v −σ (aiσ )∗ = 0 for all i ≥ 1. Thus, only those monomials in qσ (aσ1 , . . . , a−σ 1 , . . .) beginning by aσ1 give a nonzero contribution to v −σ qσ (aσ1 , . . . , a−σ 1 , . . .). Similarly, by (1.b) and (2.b), only those monomials which continue with a−σ give a nonzero 1 contribution to v −σ qσ (aσ1 , . . . , a−σ , . . .). Therefore 1 −σ σ −σ a1 )a1 )qσ,1 (aσ2 , . . . , a2−σ , . . .), v −σ qσ (aσ1 , . . . , a−σ 1 , . . .) = ((v where qσ,1 denotes a sum of monomials involving either xiτ or (xiτ )∗ for all τ ∈ {+, −}, but not both, for all 2 ≤ i ≤ d. σ Now, since the monomial xσ1 x−σ 1 . . . xd+1 does not occur in qσ , repeated application of (1) and (2) above yields v −σ qσ (aσ1 , . . . , a−σ 1 , . . .) = 0. Hence σ (((v −σ aσ1 )a−σ = 0, contrary to (1)–(3). ) . . .)a 1 d+1 ⊓ ⊔ That proves b), and a) is proved in the same way by using 3.2 instead of 3.3. 3.5 Remark Let A be an associative algebra, and consider the associative pair (A, A). If − + − (A, A) satisfies a polynomial identity p(x+ 1 , . . . , xn , x1 , . . . , xn ) of degree d, then A satisfies the algebra identity p(x1 , . . . , xn , y1 , . . . , yn ) of degree d, and therefore it is a PI algebra. Similarly, if A possesses an involution ∗, then ∗ induces an involution on (A, A), and if (A, A) satisfies a ∗-polynomial identity p(x+ , x− , (x+ )∗ , (x− )∗ ), then A satisfies an algebra ∗-identity p(x, y, x∗ , y∗ ) of the same degree. 3.6 Theorem Let A = (A+ , A− ) be a primitive associative pair, let (M+ , M− ) be a faithful irreducible right A-module, and set  = EndU A (M+ ⊕ M− ). a) If A satisfies a polynomial identity of degree 2d + 1, then for some σ = + or −, the dimension of Mσ over  is at most d. b) If A has an involution ∗, and it satisfies a ∗-polynomial identity of degree m, then M+ and M− are at most m-dimensional over . In both cases A is simple of finite capacity. Proof a) By 3.4, A has nonzero socle, and it is prime since it is primitive. By [5, 2.8], Soc(A) is a simple associative pair [4, Theorem 1] and, by [5, Theorem 2], we have Soc(A) = (F (M− , M+ ), F (M+ , M− )) ⊳ A ⊆ (Hom (M− , M+ ), Hom (M+ , M− )). Then for all n ∈ Z+ with n ≤ dim M+ and ≤ dim M− , there is a subpair of matrices Mn () = (Mn (), Mn ()) ⊆ Soc(A). This is the associative pair of the algebra Mn (), and since it is a subpair of A, it satisfies a polynomial identity of degree 2d + 1, hence Mn () satisfies an algebra identity of degree 2d + 1 by 3.5. Then 2d + 1 ≥ 2n, and we get n ≤ d, hence dim Mσ ≤ d for one of the σ = ±. On polynomial identities in associative and Jordan pairs 199 b) Since A has nonzero socle by 3.4, A is strongly prime [5, 2.8], and therefore, by [9, 3.14],  has an involution ¯, and there is a mapping g : M+ × M− →  such that (M+ , M− , g) form a pair of skew dual vector spaces over the associative division -algebra with involution (, −) (see [9, 3.11]), Soc(A) = (F (M− , M+ ), F (M+ , M− )) and the involution ∗ corresponds to either ♯ or −♯, where ♯ is the adjoint involution of (L (M− , M+ ), L (M+ , M− )). Moreover, by [9, 3.20], for any n ≤ dim (X), Soc(A) contains a ∗-subpair S isomorphic to a   full pair of matrices with involution (Mn (), Mn ()), ∗ , and again either ∗ = ♯ t and then B∗ = B or  is a field, − is the identity, ∗ = −♯, and B∗ = −Bt for all B ∈ Mn (), where t is the transpose involution.   t Suppose first that B∗ = B . Then, (Mn (), Mn ()), ∗ is the pair with involution obtained from the algebra Mn () endowed with the involution ∗. Since A satisfies a ∗-polynomial identity of degree m, so does S, hence (Mn (), ∗) satisfies an algebra ∗-polynomial identity of degree m by 3.5. Then, by a theorem of Amitsur [1], Mn () satisfies the standard identity S2m by [25, 1.4.1], hence 2n ≤ 2m, and we get n ≤ m, which yields dim M+ = dim M− ≤ m. Suppose finally that B∗ = −Bt . Since by [9, 3.20],  is now a field we write  = F. If dim F Mσ ≥ 2n for some (hence both) σ , by [9, 3.20], then Soc(A), hence A, contains a ∗-subpair S   ∗-isomorphic to a full pair of matrices with involution (M2n (F), M2n (F)), ∗ , such that, for any B = (Cij) ∈ M2n (F), B∗ = (Cij)∗ = (C∗ji ), where C ∈ M2 (F) and ∗ C =  αβ γ δ ∗ =  δ −β −γ α  for all α, β, γ , δ ∈ F.  Again, S, hence (M2n   (F), M2n (F)), ∗ , inherits the ∗-identity of A, and since (M2n (F), M2n (F)), ∗ is the ∗-pair obtained from the algebra M2n (F) endowed with some involution ∗, by 3.5, it satisfies an algebra ∗-identity of degree m, hence by [1], it satisfies the standard identity S2m . Thus we get 4n ≤ 2m by [25, 1.4.3], hence 2n ≤ m − 1 since m is odd. This implies dim F Mσ ≤ m, σ = ±. ⊓ ⊔ 3.7 Remark We point out that, as a consequence of the above proof, if A is a primitive associative pair with involution ∗, and it satisfies a ∗-polynomial identity of degree m, then there existsa division algebra   and a positive integer n such that A is isomorphic to a pair Mn (), Mn () , and Mn () satisfies the standard identity S2m . 3.8 Lemma Let A = (A+ , A− ) be an associative pair with an involution ∗ that satisfies a polynomial identity pσ (x, x∗ ) of degree m. If P is a primitive ideal of A, then A/P is simple, has finite capacity and all its local algebras satisfy the standard identity S2m . Proof If P = P∗ , the pair A/P is primitive with an involution induced by that of A, and clearly it satisfies the polynomial identity pσ (x, x∗ ) of degree m. Thus, in this case we can assume that A is primitive. By 3.7, A is then a simple pair of finite capacity 200 F. Montaner, I. Paniello   of the form Mn (), Mn () for a division algebra , and the matrix algebra Mn () satisfies the standard identity S2m . By a well known scalar extension argument (see [11, page 158]), Mn () embeds in a matrix algebra Mk (F) over a field F for some k ≤ m which also satisfies the identity S2m . If now a ∈ Aσ , then the local algebra Aa−σ = Mn ()a is (isomorphic to) a subalgebra of Mk (F)a , and it is easy to see that Mk (F)a ∼ = Mr (F), where r ≤ k ≤ m is the rank of a. Therefore the local algebra Aa−σ satisfies the standard identity S2m by Amitsur–Levitzki’s theorem [25, 1.4.1] We assume next that P  ⊇ P∗ . (Note that this case includes the possibility of A being ∗-primitive but not primitive, since then A has a primitive ideal P such that P ∩ P∗ = 0. See [25, 7.3.5].) Factoring out the ideal P ∩ P∗ we can assume that P ∩ P∗ = 0. Set I = P + P∗ = P ⊕ P∗ . Since I is a ∗-ideal of A, it inherits the ∗-polynomial identity of A. Now, if (A, ∗) satisfies the ∗-polynomial p(x+ , x− , (x+ )∗ , (x− )∗ ) of degree m, then P∗ satisfies the polynomial q = p(x+ , x− , y+ , y− ), which we can assume to be multilinear. Moreover P∗ ∼ = I/P is an ideal of the primitive pair A/P, hence it is itself a primitive pair (since any faithful irreducible A/P-module is easily seen to be a faithful irreducible I/P-module). Therefore, P∗ is a pair of finitecapacity by 3.6, and by [5, Proposition  1] has the form Hom (X, Y), Hom (Y, X) for a division algebra , and a pair of -vector spaces X, Y, with one of the dimensions dim X or dim Y finite. Note now that P∗ is an associative pair over the center Z = Z () of , and let F be a maximal subfield of . Consider now the scalar extension P∗F = P∗ ⊗ Z F. Then, (Y, X) becomes an irreducible P∗F -module, and F = EndU P∗ (Y ⊕ X) = F. Now P∗F F still satisfies the multilinear identity q of degree m, hence by 3.6, if we set m = 2d + 1 (recall that m is odd), we have that either dim F X ≤ d or dim F Y ≤ d, and P∗F is isomorphic to the pair B = Hom F (X, Y), Hom F (Y, X) . Now, if b σ ∈ Bσ , it is easy to see that the local algebra (B−σ )b σ is isomorphic to a matrix algebra Mn (F) where n is the rank of the linear transformation b σ , which is at most the minimum of dim F X and dim F Y, and hence r ≤ d < m. Therefore, every local algebra of B satisfies the standard identity S2m , and so does every local algebra of the subpair I/P = P∗ , of B. Next we claim that I/P = A/P. Indeed, since I/P has finite capacity by [15, Theorem 3(v)], there is an idempotent e = (e+ , e− ) of I/P such that (I/P)00 (e) = 0. Then I/P = (I/P)11 (e) ⊕ (I/P)10 (e) ⊕ (I/P)01 (e). But, since (I/P)ij(e) = (A/P)ij(e) for all i, j ∈ {0, 1} such that (i, j )  = (0, 0), we have A/P = I/P ⊕ (A/P)00 (e). Take now zσ ∈ (A/P)σ00 (e). By the Peirce relations (see [13, pp. 94-95]), zσ , (I/P)−σ , zσ  ⊂  (A/P)σ00 (e), (A/P)ij−σ (e), (A/P)σ00 (e) = 0, (i, j )=(0,0) which gives zσ ∈ Ann(A/P) J ((I/P)−σ ) = 0 [4, Lemma 2]. Hence (A/P)σ00 (e) = 0 and then A/P = I/P, hence A/P has finite capacity and all its local algebras satisfy the standard identity S2m . ⊓ ⊔ 3.9 Theorem Let A be an associative pair with an involution ∗. If A has a ∗-polynomial identity of degree m, then there exists a positive integer k such that every local algebra of A satisfies the polynomial identity Sk2m . Moreover, if A is semiprime, then every local algebra satisfies the standard identity S2m . On polynomial identities in associative and Jordan pairs 201 Proof We can assume that A satisfies a multilinear ∗-identity p of degree m as in 2.8. We consider first the case of a semiprime A. To deal with it, we will embed A into a semiprimitive associative pair with the same identities as A by means of its Martindale–McCrimmon embedding [20, 5.2]: Let à be the pair (Seq(A[t1 ]))[t2 ], where t1 and t2 are polynomial variables, and Seq(B) = ∞ 1 B for an associative pair B. We set E(A) = Ã/Jac( Ã), where Jac denotes the Jacobson radical. We denote by τ the composition A ⊆ à → E(A). Clearly, the involution ∗ of A extends to E(A) making τ a ∗-homomorphism of associative pairs. This is the associative version of the construction of the Martindale–McCrimmon embedding, defined in [16] for Jordan algebras, and extended to general Jordan systems in [20]. In fact, using [13, 7.9] we easily get E(A)(+) = E(A(+) ), hence by [20, Lemma 5.3], τ is injective. Moreover, E(A) satisfies the ∗-identity p, hence by 3.8, for every primitive ideal P of E(A), all local algebras of E(A)/P satisfy the standard identity S2m . Therefore, the semiprimitivity of E(A) implies that all its local algebras satisfy the identity S2m , and the same goes for A. Finally, the assertion for a general A follows by Amitsur’s argument [25, 1.6.38]. ⊓ ⊔ As an immediate application of this result we consider associative pairs with involution for which either the set of skew symmetric elements or the set or symmetric elements satisfies a polynomial identity. More generally, we can consider traces: t(r) = r + r∗ , and the set of all traces T(A, ∗) of the triple system with involution (A, ∗). 3.10 Corollary If either the set of all symmetric elements (or, more generally, the set of all traces) or the set of all skew elements of an associative pair A with involution satisfies a polynomial identity, then all local algebras of A satisfy an identity Sk2m where S2m is the standard identity. 4 Polynomial Identities in Jordan Pairs In this section we provide an affirmative answer to following conjecture which was raised in [21, 6.4]. 4.1 Conjecture Every PI-Jordan system satisfies a homotope-PI. We recall here that this result has already been proved for Jordan algebras in [20, 2.7(ii)]. Thus we will focus on Jordan pairs and triple systems, but we first recall the definition of a family of Jordan polynomials which will play for us to some extent the role of the associative standard identity. 202 F. Montaner, I. Paniello 4.2 Following [20, 2.2], we denote by Fm the family of essential polynomials in the free Jordan algebra FJ[x, y, z]  (−1)σ Vxσ (1) ,y . . . Vxσ (m+1) ,y z. Fm (x, y, z) = σ ∈Sm+1 We also write Gm (x, y, z) = Fm (x, y, z)3 . Before studying Jordan pairs, we return to algebras to obtain a sharper version of [20, 6.4(a)] for semiprime algebras: 4.3 Lemma Let A be a semiprime associative algebra. If A satisfies the standard identity S2n , then A(+) satisfies all the identities Fm for all m ≥ n2 . Proof Since A is isomorphic to a subdirect product of prime associative algebras which are homomorphic images of A and therefore also satisfy S2n , we may assume that A is prime. Then, by Posner-Rowen’s theorem, the central localization B = Z (A)−1 A of A is a simple algebra which is finite dimensional over its center, and therefore a matrix algebra over a division ring D, which still satisfies S2n . On the other hand, after a suitable scalar extension of B (for instance by a maximal subfield F of D), we obtain a matrix algebra Mk (F) which still satisfies S2n . Then k ≤ n by [25, 1.4.3], hence Mk (F) has dimension at most n2 over F, and Mk (F)(+) satisfies Fm ⊓ ⊔ for m ≥ n2 by [20, 2.3], hence A(+) also satisfies that identity. 4.4 Recall that a Jordan pair V is said to strictly satisfy a homotope polynomial identity f (y; x1 , . . . , xn ) if every scalar extension V ⊗ for a commutative associative ring ⊇ , still satisfies the identity f (y; x1 , . . . , xn ). Since the strict validity of an identity amounts to the validity of all its partial linearizations, it is easy to see that V strictly satisfies f (y) , if and only if the polynomial pair V[t] = V ⊗ [t] satisfies f (y) . 4.5 Lemma Let V be a strongly prime Jordan pair and suppose that a nonzero ideal I of V strictly satisfies some homotope polynomial identity f (y; x1 , . . . , xn ) = f (y) (x1 , . . . , xn ) for a homogeneous admissible Jordan polynomial f , then V satisfies f (y) . Proof Consider the extended central closure Ṽ = C (V)V of V, and the ideal Ĩ = C (V)I of Ṽ generated as a C (V) module by I. Take now xσ ∈ I σ and −σ −σ we have y−σ ∈ I −σ and set zσ = Qxσ y−σ . Then, for all a−σ 1 , . . . , an ∈ V −σ −σ −σ −σ σ Q y−σ f (y ; Qxσ a1 , . . . , Qxσ an ) = 0 (since Qxσ ai ∈ I for all i). Thus we have −σ −σ −σ 0 = Qxσ Q y−σ f (y−σ ; Qxσ a−σ ; a−σ 1 , . . . , Qxσ an ) = Qxσ Q y−σ Qxσ f (Qxσ y 1 , . . . , an ) = −σ −σ is PI, and thus, 0.19], hence the local algebra V Qzσ f (zσ ; a−σ , . . . , a ) by [10, σ z n 1 since zσ = Qxσ y−σ for arbitrary xσ ∈ I σ , y−σ ∈ I −σ , we have Q I σ I −σ ⊆ PI(V). Since On polynomial identities in associative and Jordan pairs 203 Q I σ I −σ  = 0 by [21, 1.3] and the primality of V, this implies PI(V)  = 0. Therefore, by [21, 5.1], Ṽ is strongly prime with nonzero socle Soc(Ṽ) = PI(C (V)V), which is a simple ideal by 2.13. Thus we get Ĩ = Soc(Ṽ) ⊆ Ĩ, and therefore Soc(Ṽ) satisfies the homotope polynomial identity f (y) , hence it has finite capacity by [20, 4.10]. This implies that Ĩ = Soc(Ṽ) = Ṽ using a complete idempotent as in 3.8, and therefore Ṽ, hence V, satisfies f (y) . ⊓ ⊔ 4.6 Proposition Let V be a nondegenerate Jordan pair. If V satisfies a polynomial identity of degree m, then all its local algebras satisfy the identities Fk (x, y, z) for all k ≥ m2 , hence V satisfies the homotope polynomial identity Gk (t; x, y, z) for all k ≥ m2 . Proof Since any nondegenerate Jordan pair is a subdirect product of strongly prime Jordan pairs, it clearly suffices to consider the case of a strongly prime V. In that case, by [2, 4.3], either there exists a scalar extension Ṽ of V which is Clifford, bi-Cayley or Albert, or V consists of hermitian elements: V has a nonzero ideal I = H0 (A, ∗) which is an ample subpair of a ∗-prime associative pair A with involution ∗, and V ⊆ H(Q(A), ∗), where Q(A) is the Martindale pair of symmetric quotients of A. We consider first the hermitian case. Note that we can assume that V satisfies a multilinear polynomial identity p of degree m. Consider the polynomial extension I[t] = I ⊗ [t]. It is straightforward that I[t] = H0 (A[t], ∗) is an ample subpair of the ∗-prime associative pair A[t], and it still satisfies the polynomial identity p. By 3.9, every local algebra of the semiprime pair A[t] satisfies the standard identity S2m , and since these are again semiprime, every local algebra of A[t](+) satisfies all the identities Fm for all m ≥ n2 by 4.3, and so does every local algebra of I[t]. Thus I strictly satisfies all the identities Fm (t; x, y, z) for all m ≥ n2 by 4.4. Thus V satisfies (t) all the identities Fm for all m ≥ n2 by 4.5. Now if V is bi-Cayley or of Albert type, then all its local algebras are at most 27dimensional over their centroids, and if V is of Clifford type, then its local algebras are generically algebraic of degree 2 over their centroids. Therefore they all satisfy Fk (x, y, z) for k ≥ 27. Since m ≥ 3, all local algebras of V satisfy Fk for k ≥ m2 . ⊓ ⊔ 4.7 Theorem Let J be a Jordan system. If J satisfies a polynomial identity of degree m, then J satisfies the homotope polynomial identity Fkl (t; x, y, z) = Fk (t; x, y, z)(l,t) for all k ≥ m2 and some l ≥ 3. Proof For Jordan pairs, this follows from 4.6 by Amitsur’s argument [25, 1.6.38]. The result for Jordan triple systems follows from that result by using the double pair V(J) attached to a Jordan triple system J. ⊓ ⊔ As a consequence of that theorem we can improve the Jordan analogues of Kaplansky’s and Posner-Rowen theorems obtained in [20–22]. 204 F. Montaner, I. Paniello 4.8 Theorem Let V be a Jordan pair, and suppose that V satisfies a polynomial identity, then: a) If V is primitive, then it is simple of finite capacity. b) If V is strongly prime, then its central closure Ŵ(V)−1 V is simple of finite capacity. c) The McCrimmon radical and the properly nilpotent radical of V coincide. Proof Since by 4.7 every PI Jordan system satisfies a homotope polynomial identity, (a) follows from [20, 4.10(ii)], (b) from [22, 4.3] and, finally, the McCrimmon radical ⊓ ⊔ and the properly nilpotent radical of V coincide by [20, 6.3(b)]. 4.9 Remark Part c) of 4.8 was first proved by Zelmanov in [26, Theorem 3]. This raises the question of whether the present proof is independent of that result, which was used in [26] to prove its prime dichotomy theorem (every strongly prime Jordan system is either i-special or an Albert form), which in turn is needed in the proof of 4.6. However a careful reading of [26] reveals that the only result that is needed is [20, 6.3(b)], which does not require the classification theorem of strongly prime Jordan systems. 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