Local and Subquotient Inheritance of the Heart of a Jordan System

José A. Anquela, Teresa Cortés, Esther García

Dedicated to Prof. Dr. Kurt Meyberg on the occasion of his 65th birthday

Abstract: We establish the inheritance of the heart by local algebras and subquotients of Jordan and associative systems.

Introduction

For semiprime associative or nondegenerate Jordan systems, having a nonzero heart turns out to be a natural condition to guarantee the nice behaviour of the system. For instance, the proof of the Jordan algebra version of Kaplansky’s Theorem [8, 1.2] makes use of this property, which can be considered as a strong version of primeness.

The heart of a Jordan system is a particular and relevant case of minimal ideal. Simplicity of minimal ideals has been studied among other authors by Ng Seong Nam and Kevin McCrimmon [16] and is still an open problem in its whole generality for Jordan systems. In [6] it is obtained that the heart of a nondegenerate Jordan system is simple when nonzero.

On the other hand, local algebras of Jordan and associative systems, introduced by Kurt Meyberg [14], are one of the most useful tools to transfer back and forth regularity conditions between the categories of algebras and pairs and triple systems. Linked to the study of local algebras, one can consider subquotients of a Jordan pair determined by an inner ideal [11], which can be seen as a generalization of that notion.

In this paper we study how the heart is inherited by local algebras and subquotients. The main tools are, on the one hand, the simplicity of the heart obtained in

José A. Anquela, Teresa Cortés, Esther García Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain e-mail addresses: anque@pinon.ccu.uniovi.es, cortes@pinon.ccu.uniovi.es,

egg@pinon.ccu.uniovi.es

2000 Math. Subj. Class.: 17C10, 17C20 [6] and, on the other hand, the results on inheritance of simplicity by local algebras and subquotients established in [4].

Concerning the question about local inheritance of the heart, studied in the first section, we remark that a positive answer can only be obtained for algebras and pairs. Triple systems have a traditional worse behaviour concerning the inheritance of regularity conditions by local algebras, and the inheritance of the heart is not an exception. The section includes results on local-to-global inheritance as well as associative analogues with and without involution. The nature of the problem involves the so-called “generalized” local algebras of ideals. Similarly, in the second section, devoted to subquotient inheritance of the heart, a study of the interaction of subquotients and ideals is needed.

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 also consider involutions on associative systems and, when dealing with associative pairs, involutions will be assumed of polarized type. The reader is referred to [1, 9, 10, 13] for basic results, notation and terminology, though we will stress some definitions and basic notions. The identities JPx listed in [10] will be quoted with their original numbering without explicit reference to [10].

—Given a Jordan algebra J, its products will be denoted x^{2}, U_{x}y, for x, y ∈ J. They are quadratic in x and linear in y and have linearizations denoted x∘y, U_{x,z}y = {x, y, z} = V_{x,y}z, respectively.

—For a Jordan pair V = (V^{+}, V^{−}) we will denote the products by Q_{x}y, for any x ∈ V^{σ}, y ∈ V^{−σ}, σ = ±, with linearizations denoted by Q_{x,z}y = {x, y, z} = D_{x,y}z.

—A Jordan triple system T is given by its products P_{x}y, for any x, y ∈ T, with linearizations denoted by P_{x,z}y = {x, y, z} = L_{x,y}z.

0.2 We will deal with the usual notions of regularity in Jordan systems: simplicity, primeness, nondegeneracy, whose definitions and basic properties can be found in the general references listed above. Concerning nondegeneracy we will stress a well known fact that will be used several times in the sequel:

If J is a nondegenerate Jordan system (algebra, triple system or pair) then every nonzero ideal of J is nondegenerate (cf. [12, 3.5]).

0.3 A Jordan algebra gives rise to a Jordan triple system by simply forgetting the squaring and letting P = U. By doubling any Jordan triple system T one obtains the double Jordan pair V (T) = (T, T) with products Q_{x}y = P_{x}y, for any x, y ∈ T. From a Jordan pair V = (V^{+}, V^{−}) one can get a (polarized) Jordan triple system T (V) = V^{+} ⊕ V^{−} by defining P_{x^{+}⊕x^{−}}(y^{+} ⊕ y^{−}) = Q_{x^{+}}y^{−} ⊕ Q_{x^{−}}y^{+} [10, 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 Simplicity inheritance through the functor T ( ) [1, p. 230]: A Jordan pair V is simple if and only if T (V) is simple.

0.5 The heart Heart(S) of an associative or Jordan system S is the intersection of all nonzero ideals of S. For an associative system S with involution ∗, the ∗-heart ∗-Heart(S) of S is the intersection of all nonzero ∗-ideals of S.

We will need the following properties of the heart:

0.6 Simple ideals turn out to be the hearts of sufficiently regular systems as shown in [6, 0.9]:

(i) Let J be a strongly prime Jordan (resp. prime associative) algebra, triple system or pair. If L is an ideal of J which is a simple system, then L = Heart(J).

(ii) Let R be a ∗-prime associative algebra, triple system or pair. If L is a ∗-ideal which is a ∗-simple system, then L = ∗-Heart(R).

0.7 If J is a nondegenerate Jordan algebra, triple system or pair and Heart(J) ≠ 0, then J is prime (hence strongly prime) [6, 2.3, 3.3, 3.10].

0.8 If R is a semiprime associative algebra, triple system or pair (resp. a semiprime associative algebra, triple system or pair with involution ∗) and Heart(R) ≠ 0 (resp. ∗-Heart(R) ≠ 0), then R is prime (resp. ∗-prime) [6, 2.4, 3.4, 3.11].

0.9 If J is a nondegenerate Jordan algebra, triple system or pair, then the heart Heart(J) of J is either simple or zero [6, 2.6, 3.6, 3.13].

0.10 The heart (resp. the ∗-heart) of a semiprime associative algebra, triple system or pair (resp. a semiprime associative algebra, triple system or pair with involution ∗) R is simple (resp. ∗-simple) when it is nonzero [6, 2.1, 3.1, 3.8].

1. Local Algebras

1.1 Local algebras of Jordan and associative 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}. In R^{σ(a)} one can find the ideal

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

and the quotient

R_{a}^{σ} = R^{σ(a)}/ Ker a

is called the local algebra of R at a.

When the associative pair is endowed with an involution, we will usually take a symmetric element a, so that R^{σ(a)} and R_{a}^{σ} inherit the involution.

—Given a Jordan pair V = (V^{+}, V^{−}) and a ∈ V^{−σ}, the Φ-module V^{σ} becomes a Jordan algebra, denoted V^{σ(a)} and called the a-homotope of V, with products

x^{(2,a)} = Q_{x}a, U_{x}^{(a)}y = Q_{x}Q_{a}y

for any x, y ∈ V^{σ}. In V^{σ(a)} one can find the ideal

Ker_{V} a = Ker a = {x ∈ V^{σ} | Q_{a}x = Q_{a}Q_{x}a = 0}

and the quotient

V_{a}^{σ} = V^{σ(a)}/ Ker a

is local algebra of V at a. When V is nondegenerate or special, Ker_{V} a = {x ∈ V^{σ} | Q_{a}x = 0}.

—Homotopes and local algebras of a Jordan (resp. associative) algebra or triple system J at an element a are simply those of the Jordan (resp. associative) pair V (J).

The above notions are compatible with the functors V ( ) and T ( ) (cf. [3, 0.5]).

1.2 Let I be an ideal of V, where V is an associative or Jordan pair. For any a ∈ V^{−σ}, I^{σ} is an ideal of V^{σ(a)} which we will call the homotope of I at a, and denote I^{σ(a)}, even if a ∉ I^{−σ}. On the other hand, Ker_{V} a ∩ I^{σ} is an ideal of I^{σ(a)} which will be denoted Ker_{I} a. The quotient I^{σ(a)}/ Ker_{I} a will be called the local algebra of I at a and denoted I_{a}^{σ}. Notice that I_{a}^{σ} is naturally isomorphic to the ideal (I^{σ} + Ker_{V} a)/ Ker_{V} a of V_{a}^{σ}.

When dealing with Jordan pairs, if I is nondegenerate (for example, by (0.2), when V is nondegenerate), Ker_{I} a = {x ∈ I^{σ} | Q_{a}x = 0} [if Q_{a}x = 0, then Q_{Q_{a}Q_{x}a}I^{σ} = Q_{a}Q_{x}Q_{a}Q_{x}Q_{a}I^{σ} = Q_{Q_{a}x}Q_{x}Q_{a}I^{σ} = 0; thus Q_{a}Q_{x}a ∈ I^{−σ} is an absolute zero divisor of I and Q_{a}Q_{x}a = 0 by nondegeneracy of I].

When the associative pair is endowed with an involution ∗ and I is a ∗-ideal, we will usually take a symmetric element a, so that I^{σ(a)} and I_{a}^{σ} inherit the involution.

Given an ideal I of J, where J is an associative or Jordan algebra or triple system, homotopes and local algebras of I at an element a ∈ J are simply those of the ideal V (I) in V (J).

We will study how local algebras interact with hearts. We start by extending to the “generalized” local algebras considered in (1.2) the results of inheritance of strong primeness [7, 3.2] and simplicity [4, 2.5].

1.3 Proposition. Let V be a Jordan (resp. associative) pair and I be an ideal of V. If I is strongly prime (resp. prime) then, for any a ∈ V^{−σ} (σ = ±), I_{a}^{σ} is either strongly prime (resp. prime) or zero.

(Allowing the zero algebra to be said strongly prime (resp. prime) would simplify the above statement)

Proof: Put, for example, σ = + and assume that I_{a}^{+} ≠ 0.

(I) Assume first that V is a Jordan pair. To prove that I_{a}^{+} is strongly prime we just need to show that it is elementally prime [7, 1.13].

We will denote J = I_{a}^{+}, and, for any x ∈ I^{+}, we will write x̄ = x + Ker_{I} a ∈ J. Let x̄, ȳ ∈ J such that Ux̄U_{J}UȳJ = 0. Thus Q_{x}Q_{a}Q_{I^{+}}Q_{a}Q_{y}Q_{a}I^{+} ⊆ Ker_{I} a and

Q_{Q_{a}x}Q_{I^{+}}Q_{Q_{a}y}I^{+} = Q_{a}Q_{x}Q_{a}Q_{I^{+}}Q_{a}Q_{y}Q_{a}I^{+} = 0.

Since I is strongly prime, we have that Q_{a}x = 0 or Q_{a}y = 0 by [7, 1.10], which implies x ∈ Ker_{I} a or y ∈ Ker_{I} a, i.e., x̄ = 0 or ȳ = 0.

(II) Assume that V is an associative pair. We will denote R = I_{a}^{+}, and, for any x ∈ I^{+}, we will write x̄ = x + Ker_{I} a ∈ R.

Notice that R is semiprime: x̄Rx̄ = 0 implies xaI^{+}ax ⊆ Ker_{I} a, i.e., axaI^{+}axa = 0; hence axa = 0 by semiprimeness of I, which means x ∈ Ker_{I} a, i.e., x̄ = 0.

Let A = A_{1}/ Ker_{I} a, B = B_{1}/ Ker_{I} a be nonzero ideals of R, which implies that A_{1}, B_{1} are ideals of I^{+(a)} such that aA_{1}a ≠ 0 and aB_{1}a ≠ 0. By primeness of I, aA_{1}aI^{+}aB_{1}a ≠ 0, which means that A_{1}aI^{+}aB_{1} is not contained in Ker_{I} a. Hence 0 ≠ (A_{1}aI^{+}aB_{1} + Ker_{I} a)/ Ker_{I} a is contained in A ∩ B, which, with semiprimeness of R, shows that R is prime.

By replacing ideals by ∗-ideals and primeness by ∗-primeness in part (II) of the proof of (1.3), we obtain a version of this result for associative pairs with involution (notice that, unlike in (1.3)(I), we have avoided using an associative elemental characterization of primeness in (1.3)(II) to be able to use its proof here):

1.4 Proposition. Let R be an associative pair with involution ∗ and I be a ∗-ideal of R. If I is ∗-prime then, for any a ∈ H(R^{−σ}, ∗) (σ = ±), I_{a}^{σ} is either ∗-prime or zero.

1.5 Proposition. Let V be a Jordan (resp. associative) pair and I be an ideal of V. If I is simple then, for any a ∈ V^{−σ} (σ = ±), I_{a}^{σ} is either simple or zero.

Proof: Put, for example, σ = + and assume that I_{a}^{+} ≠ 0.

(I) Assume that V is a Jordan pair. We will denote J = I_{a}^{+}, and, for any x ∈ I^{+}, we will write x̄ = x + Ker_{I} a ∈ J.

We claim that J is not trivial: Indeed, by [4, 2.4 (ii)], I is strongly prime, hence J is strongly prime by (1.3).

Now, to show that J is simple we just follow the proof of [4, 2.5]: Let L be a nonzero ideal of J, i.e., L = L_{1}/ Ker_{I} a, where L_{1} is an ideal of I^{+(a)} strictly containing Ker_{I} a. Thus, there exists x ∈ L_{1} such that Q_{a}x ≠ 0 (see the description of Ker_{I} a in (1.2)).

Define K^{+} := Q_{x}Q_{a}I^{+}, K^{−} := Q_{a}L_{1}, which are inner ideals of I:
Q_{K^{+}}I^{−} = Q_{Q_{x}Q_{a}I^{+}}I^{−} = Q_{x}Q_{a}Q_{I^{+}}Q_{a}Q_{x}I^{−} (by JP3) ⊆ Q_{x}Q_{a}I^{+} = K^{+},
Q_{K^{−}}I^{+} = Q_{Q_{a}L_{1}}I^{+} = Q_{a}Q_{L_{1}}Q_{a}I^{+} (by JP3) = Q_{a}U_{L_{1}}^{(a)}I^{+} ⊆ Q_{a}L_{1} = K^{−}.
Moreover Q_{K^{+}}K^{−} ≠ 0: indeed, suppose
0 = Q_{K^{+}}K^{−} = Q_{Q_{x}Q_{a}I^{+}}Q_{a}L_{1} = U_{Q_{x}Q_{a}I^{+}}^{(a)}L_{1}.  (i)

Since J is strongly prime, we have Ann_{J}(L) = 0 by [12, 1.6]. But (i) implies (Q_{x}Q_{a}I^{+} + Ker_{I} a)/ Ker_{I} a is contained in the annihilator Ann_{J}(L) of L in J (cf. [12, 1.7]). We then have (Q_{x}Q_{a}I^{+} + Ker_{I} a)/ Ker a = 0, hence Q_{a}Q_{x}Q_{a}I^{+} = 0, i.e., Q_{Q_{a}x}I^{+} = 0 where 0 ≠ Q_{a}x ∈ I^{−}, which is impossible by nondegeneracy of I.

Let K = K^{+} ⊕ K^{−}, which is clearly an inner ideal of T (I). Moreover, P_{K}K = Q_{K^{+}}K^{−} ⊕ Q_{K^{−}}K^{+} ≠ 0. Since T (I) is simple by (0.4)(iii), we have that the ideal of T (I) generated by P_{K}K is the whole T (I). By [2, 4.13], T (I) is locally nilpotent mod K + P_{T (I)}K, hence T (I) = K + P_{T (I)}K by [4, 2.3]. But

K + P_{T (I)}K = (K^{+} + Q_{I^{+}}K^{−}) ⊕ (K^{−} + Q_{I^{−}}K^{+}),

hence

I^{+} = K^{+} + Q_{I^{+}}K^{−} = Q_{x}Q_{a}I^{+} + Q_{I^{+}}Q_{a}L_{1} = U_{x}^{(a)}I^{+} + U_{I^{+}}^{(a)}L_{1} ⊆ L_{1}

since x ∈ L_{1} and L_{1} is an ideal of I^{+(a)}. Thus L_{1} = I^{+}, hence L = J, and we have shown that J is simple.

(II) Assume that V is an associative pair. We will denote R = I_{a}^{+}. Notice that R is not trivial since it is prime by (1.3).

If 0 ≠ L is an ideal of R, L = L_{1}/ Ker_{I} a, where L_{1} is an ideal of I^{+(a)}. Now, L ≠ 0 implies that there exists y ∈ L_{1} such that aya ≠ 0. By simplicity, I equals the ideal (I^{+}ayaI^{+}, Φaya + I^{−}I^{+}aya + ayaI^{+}I^{−} + I^{−}I^{+}ayaI^{+}I^{−}) of I generated by aya, hence I^{+} = I^{+}ayaI^{+}, but I^{+}ayaI^{+} = I^{+} ·_{a} y ·_{a} I^{+} ⊆ L_{1} and I^{+} = L_{1}, i.e., R = L.

Again by replacing ideals by ∗-ideals in part (II) of the proof of (1.5), we obtain a version of this result for associative pairs with involution:

1.6 Proposition. Let R be an associative pair with involution ∗ and I be a ∗-ideal of R. If I is ∗-simple then, for any a ∈ H(R^{−σ}, ∗) (σ = ±), I_{a}^{σ} is either ∗-simple or zero.

Now we have the ingredients needed to study the heart of a Jordan pair in terms of its local algebras.

1.7 Theorem. Let V be a nonzero Jordan (resp. associative) pair. Then the following are equivalent:

(i) V is nondegenerate (resp. semiprime) and Heart(V) ≠ 0.

(ii) Heart(V_{a}^{σ}) ≠ 0 for every 0 ≠ a ∈ V^{−σ} (σ = ±).

(iii) V is nondegenerate (resp. semiprime) and Heart(V_{a}^{+}) ≠ 0 for every 0 ≠ a ∈ V^{−}.

(iv) V is nondegenerate (resp. semiprime) and Heart(V_{a}^{−}) ≠ 0 for every 0 ≠ a ∈ V^{+}.

(v) V is strongly prime (resp. prime) and there exists 0 ≠ a ∈ V^{−} such that Heart(V_{a}^{+}) ≠ 0.

(vi) V is strongly prime (resp. prime) and there exists 0 ≠ a ∈ V^{+} such that Heart(V_{a}^{−}) ≠ 0.

Moreover, if any of the above conditions holds, then

Heart(V_{a}^{σ}) ≅ Heart(V)_{a}^{σ}

for any 0 ≠ a ∈ V^{−σ} (σ = ±).

Proof: We will write the proof in the case when V is a Jordan pair. The proof of the associative case follows by replacing the Jordan notions and results by their corresponding associative versions.

(i) ⇒ (ii): By (0.7), V is strongly prime. Let 0 ≠ a ∈ V^{−}, for example. Put I = Heart(V). By [3, 4.1], Q_{a}I^{+} ≠ 0, which implies I^{+} is not contained in Ker_{I} a, i.e., I_{a}^{+} ≠ 0. Thus I_{a}^{+} is simple by (1.5), since I is simple by (0.9). Moreover, I_{a}^{+} is isomorphic to an ideal L of V_{a}^{+}. Thus L is a simple ideal of V_{a}^{+} which is strongly prime by [7, 3.2]. By (0.6)(i), L is the heart of V_{a}^{+}. Notice that we have established

Heart(V_{a}^{σ}) ≅ Heart(V)_{a}^{σ} ≠ 0.

(ii) ⇒ (iii): Since, in particular, V_{a}^{σ} ≠ 0 for every 0 ≠ a ∈ V^{−σ} (σ = ±), we have that V is nondegenerate.

(iii) ⇒ (v): In particular, we have that V is nondegenerate and V_{a}^{+} ≠ 0 for every 0 ≠ a ∈ V^{−}. Moreover, V_{a}^{+} is nondegenerate by [2, 3.1 (i)], which implies that V_{a}^{+} is strongly prime by (0.7). Thus, V is strongly prime by [7, 3.2].

[To extend the above argument to associative systems one needs an associative analogue of [7, 3.2] which follows from [7, 3.2] by taking symmetrizations of the associative objects involved, using [1, 1.6; 5, 1.2(ii), 1.7(i)]]

(v) ⇒ (i): Let L̄ = L/ Ker_{V} a = Heart(V_{a}^{+}), where L is an ideal of V^{+(a)} strictly containing Ker_{V} a, i.e., Q_{a}L ≠ 0. For any nonzero ideal A of V, Q_{a}A^{+} ≠ 0 by [3, 4.1] since V is strongly prime. Thus, (A^{+} + Ker_{V} a)/ Ker_{V} a is a nonzero ideal of V_{a}^{+}, hence Heart(V_{a}^{+}) ⊆ (A^{+} + Ker_{V} a)/ Ker_{V} a, which implies L ⊆ A^{+} + Ker_{V} a. Therefore Q_{a}L ⊆ Q_{a}(A^{+} + Ker_{V} a) = Q_{a}A^{+} ⊆ A^{−}. We have shown that 0 ≠ Q_{a}L ⊆ Heart(V)^{−}.

By (+/−)-symmetry, the implications (i) ⇒ (ii) ⇒ (iv) ⇒ (vi) ⇒ (i) are also true.

We also have the version of (1.7) for associative pairs with involution.

1.8 Theorem. Let R be an associative pair with involution ∗.

(i) If R is semiprime and ∗-Heart(R) ≠ 0, then ∗-Heart(R_{a}^{σ}) ≠ 0 for every 0 ≠ a ∈ H(R^{−σ}, ∗) (σ = ±). Moreover,

∗-Heart(R_{a}^{σ}) ≅ ∗-Heart(R)_{a}^{σ}

for any a ∈ H(R^{−σ}, ∗) (σ = ±).

(ii) If R is ∗-prime and there exists 0 ≠ a ∈ H(R^{−}, ∗) such that ∗-Heart(R_{a}^{+}) ≠ 0, then ∗-Heart(R) ≠ 0.

(iii) If R is ∗-prime and there exists 0 ≠ a ∈ H(R^{+}, ∗) such that ∗-Heart(R_{a}^{−}) ≠ 0, then ∗-Heart(R) ≠ 0.

Proof: (i) By (0.8), R is ∗-prime. Let 0 ≠ a ∈ H(R^{−}, ∗), for example. Put I = ∗-Heart(R). By ∗-primeness of R, aI^{+}a ≠ 0, which implies I^{+} is not contained in Ker_{I} a, i.e., I_{a}^{+} ≠ 0. Thus I_{a}^{+} is ∗-simple by (1.6), since I is ∗-simple by (0.10). Moreover, I_{a}^{+} is isomorphic to a ∗-ideal L of R_{a}^{+}. Thus L is a ∗-simple ideal of R_{a}^{+}, which is ∗-prime since R is so. By (0.6)(ii), L is the ∗-heart of R_{a}^{+} and we also have

∗-Heart(R_{a}^{σ}) ≅ ∗-Heart(R)_{a}^{σ} ≠ 0.

(ii) Let L̄ = L/ Ker_{R} a = ∗- Heart(R_{a}^{+}), where L is a ∗-ideal of R^{+(a)} strictly containing Ker_{R} a, i.e., aLa ≠ 0. For any nonzero ∗-ideal A of R, aA^{+}a ≠ 0 by ∗-primeness of R. Thus, (A^{+} + Ker_{R} a)/ Ker_{R} a is a nonzero ∗-ideal of R_{a}^{+}, hence ∗-Heart(R_{a}^{+}) ⊆ (A^{+} + Ker_{R} a)/ Ker_{R} a, which implies L ⊆ A^{+} + Ker_{R} a. Therefore aLa ⊆ a(A^{+} + Ker_{R} a)a = aA^{+}a ⊆ A^{−}. We have shown that 0 ≠ aLa ⊆ ∗- Heart(R)^{−}.

(iii) follows from (ii) by (+/−)-symmetry.

Applying [6, 4.4(i); 7, 1.12] and the (straightforward) associative version of [7, 1,12] to (1.7) yields:

1.9 Corollary. Let J be a nonzero Jordan (resp. associative) algebra. Then the following are equivalent:

(i) J is nondegenerate (resp. semiprime) and Heart(J) ≠ 0.

(ii) Heart(J_{a}) ≠ 0 for every 0 ≠ a ∈ J.

(iii) J is strongly prime (resp. prime) and there exists 0 ≠ a ∈ J such that Heart(J_{a}) ≠ 0.

Moreover, if any of the above conditions holds, then

Heart(J_{a}) ≅ Heart(J)_{a}

for any 0 ≠ a ∈ J.

The following result follows from (1.8), using [6, 4.4(ii)] and the (straightforward) ∗-associative version of [7, 1.12].

1.10 Corollary. Let R be an associative algebra with involution ∗.

(i) If R is semiprime and ∗-Heart(R) ≠ 0, then ∗-Heart(R_{a}) ≠ 0 for every 0 ≠ a ∈

H(R, ∗). Moreover,

∗-Heart(R_{a}) ≅ ∗-Heart(R)_{a}

for any a ∈ H(R, ∗).

(ii) If R is ∗-prime and there exists 0 ≠ a ∈ H(R, ∗) such that ∗-Heart(R_{a}) ≠ 0, then ∗-Heart(R) ≠ 0.

1.11 Remark: (i) Primeness of V is needed in (1.7)(v)(vi) as shown by the following example: Let V be the direct sum V_{1} + V_{2}, where V_{1} is simple and V_{2} is nondegenerate. Since V_{1} is also nondegenerate by [4, 2.4 (ii)], V is nondegenerate. Clearly the heart of V is zero since (V_{1} + 0) ∩ (0 + V_{2}) = 0 but, for any 0 ≠ a ∈ V_{1}^{−}, V_{a+0}^{+} is isomorphic to (V_{1})_{a}^{+}, which is simple by [4, 2.5], hence Heart(V_{a+0}^{+}) = V_{a+0}^{+} ≠ 0 (obviously there exists some 0 ≠ a ∈ V_{1}^{−} since, by simplicity of V_{1}, V_{1}^{−} ≠ 0).

(ii) There is no triple analogue of (1.7) since simplicity of triple systems is not preserved by taking local algebras [4, 2.6].

2. Subquotients

2.1 Let V be a Jordan pair, M ⊆ V^{+} be an inner ideal of V. The set

Ker_{V} M = Ker M = {x ∈ V^{−} | Q_{M}x = Q_{M}Q_{x}M = 0}

gives rise to the ideal (0, Ker M) of the subpair (M, V^{−}) of V. The quotient

(M, V^{−})/(0, Ker M) = (M, V^{−}/ Ker M)

is called the subquotient of V determined by M [11, 1.5]. Similarly, one can consider subquotients of V determined by an inner ideal M ⊆ V^{−} of V. When V is nondegenerate or special, Ker_{V} M = {x ∈ V^{−σ} | Q_{M}x = 0} (cf. [11, 2.4]).

To study the interaction of hearts and subquotients, the following considerations will be needed.

2.2 Let V be a nondegenerate Jordan pair, M be a nonzero inner ideal of V, M ⊆ V^{σ}, and I be a nonzero ideal of V. Clearly M ∩ I^{σ} is an inner ideal of I.

Put σ = +, for example. It is clear that I ∩ (M, V^{−}) = (M ∩ I^{+}, I^{−}) is an ideal of the subpair (M, V^{−}) of V. Hence

((M ∩ I^{+}, I^{−}) + (0, Ker_{V} M))/(0, Ker_{V} M) = (M ∩ I^{+}, (I^{−} + Ker_{V} M)/ Ker_{V} M) is an ideal of the subquotient of V determined by M, clearly isomorphic to

(M ∩ I^{+}, I^{−}/(I^{−} ∩ Ker_{V} M)).

We claim that

(i) I^{−} ∩ Ker_{V} M = Ker_{I}(M ∩ I^{+}).

Indeed, since I^{−} ∩ Ker_{V} M ⊆ Ker_{I}(M ∩ I^{+}), we just need to show Ker_{I}(M ∩ I^{+}) ⊆ Ker_{V} M: Let x ∈ Ker_{I}(M ∩ I^{+}), m ∈ M, y ∈ I^{−}. Then

Q_{Q_{m}Q_{x}Q_{m}y}I^{−} = Q_{m}Q_{x}Q_{m}Q_{y}Q_{m}Q_{x}Q_{m}I^{−} (by JP3) =
Q_{m}Q_{x}Q_{Q_{m}y}Q_{x}Q_{m}I^{−} (by JP3) ⊆ Q_{m}Q_{x}Q_{M∩I^{+}}Q_{x}(M ∩ I^{+}) = 0

since x ∈ Ker_{I}(M ∩ I^{+}). By (0.2), I is nondegenerate, hence

0 = Q_{m}Q_{x}Q_{m}y = Q_{Q_{m}x}y.

Thus we have Q_{Q_{m}x}I^{−} = 0, which implies Q_{m}x = 0, again by nondegeneracy of I, i.e, x ∈ Ker_{V} M.

(ii) We have shown that the subquotient of I determined by M ∩I^{σ} is isomorphic to an ideal of the subquotient of V determined by M.

(iii) Moreover, if V is strongly prime, M ∩ I^{σ} ≠ 0 since it contains Q_{M}I^{−σ}, which is nonzero by [3, 4.1]. Thus the subquotient of I determined by M ∩ I^{σ} is isomorphic to a nonzero ideal of the subquotient of V determined by M.

We will also need the following generalization of (0.6) for subideals.

2.3 Lemma. (i) Let J be a strongly prime Jordan (resp. prime associative) algebra, triple system or pair such that Heart(J) ≠ 0. If L is a subideal of J which is a simple system, then L = Heart(J).

(ii)Let R be a ∗-prime associative algebra, triple system or pair such that ∗- Heart(R) ≠ 0. If L is a ∗-subideal which is a ∗-simple system, then L = ∗-Heart(R).

Proof: We will prove (i) for Jordan pairs. The remaining cases of (i) and (ii) can be proved analogously.

Let L_{1}, …, L_{n} be pairs of submodules of J such that L = L_{n} is an ideal of L_{n−1}, …, L_{1} is an ideal of J. We remark that L ≠ 0 since L is simple, hence L_{i} ≠ 0 for all i. Let I = Heart(J).

By [12, 2.5], L_{1}, …, L_{n} are strongly prime Jordan pairs. Since L_{1} is a nonzero ideal of J, I ⊆ L_{1} and I is an ideal of L_{1}. Moreover, I is simple by (0.9), hence I = Heart(L_{1}) by (0.6). Inductively one can show that I = Heart(L_{i}) for any i = 1, …, n. In particular, I = Heart(L_{n}) = Heart(L), but Heart(L) = L by simplicity of L.

2.4 Theorem. Let V be a nondegenerate Jordan pair.

(i) Let M be a nonzero inner ideal of V, M ⊆ V^{σ}, S be the subquotient of V determined by M and I be the heart of V. If I ≠ 0, then the heart Heart(S) of S is nonzero and it is given by

Heart(S)^{σ} = Q_{M∩I^{σ}}I^{−σ},  Heart(S)^{−σ} = (I^{−σ} + Ker_{V} M)/ Ker_{V} M (1)

and it is isomorphic to the heart T of the subquotient of I determined by M ∩I^{σ}, where

T^{σ} = Q_{M∩I^{σ}}I^{−σ},  T^{−σ} = I^{−σ}/ Ker_{I}(M ∩ I^{σ}).  (2)

(ii) If V is strongly prime and some subquotient has a nonzero heart, then the heart of V is nonzero.

Proof: (i) By (0.9), I is simple, hence (2) and the fact that T is simple (hence nonzero) follow from [4, 5.1], since M ∩I^{σ} ≠ 0 (use (2.2)(iii) since V is strongly prime by (0.7)). Let U be given by

U^{σ} = Q_{M∩I^{σ}}I^{−σ}, U^{−σ} = (I^{−σ} + Ker_{V} M)/ Ker_{V} M,

and W by

W^{σ} = M ∩ I^{σ}, W^{−σ} = (I^{−σ} + Ker_{V} M)/ Ker_{V} M.

Now W is an ideal of S (see (2.2)), and U is clearly an ideal of W since it contains W^{3}. Moreover U is readily seen to be isomorphic to T by (2.2)(i). We have that

(3) U is a simple subideal of S.

(4) S is strongly prime by [7, 3.2] since V is strongly prime by (0.7).

(5) Heart(S) ≠ 0: Take an arbitrary local algebra J of S at some nonzero element of S^{σ} = M. By [7, 3.1] J is isomorphic to a local algebra of V, and has a nonzero heart by (1.7). Hence S has a nonzero heart by (1.7) since it is strongly prime.

Now, (3), (4) and (5) imply U = Heart(S) by (2.3).

(ii) Let M be a nonzero inner ideal of V, M ⊆ V^{σ}, and S be the subquotient of V determined by M. If the heart of S is nonzero, then every local algebra J of S has a nonzero heart by (1.7). But for a local algebra J at a nonzero element in S^{σ}, J is isomorphic to a local algebra of V [7, 3.1], hence V has a nonzero heart by (1.7).

2.5 Corollary. Let V be a regular Jordan pair. Let M be a nonzero inner ideal of V, M ⊆ V^{σ}, S be the subquotient of V determined by M, and I be the heart of V. If I ≠ 0, then the heart Heart(S) of S is nonzero and it is given by

Heart(S)^{σ} = M ∩ I^{σ},  Heart(S)^{−σ} = (I^{−σ} + Ker_{V} M)/ Ker_{V} M  (1)

and it is isomorphic to the the subquotient of I determined by M ∩ I^{σ}.

Proof: Every regular Jordan pair is nondegenerate, thus we just need to apply (2.4)(i) once we prove that M ∩ I^{σ} = Q_{M∩I^{σ}}I^{−σ}. Since M ∩ I^{σ} is an inner ideal of I (cf. (2.2)), we just need to prove M ∩ I^{σ} ⊆ Q_{M∩I^{σ}}I^{−σ}: Let m ∈ M ∩ I^{σ}. By regularity of V, there exists x ∈ V^{−σ} such that m = Q_{m}x. Hence,

m = Q_{m}x = Q_{Q_{m}x}x = Q_{m}Q_{x}Q_{m}x (by JP3) ∈ Q_{M∩I^{σ}}I^{−σ}

since Q_{x}Q_{m}x ∈ I^{−σ}.

Acknowledgements: This paper has been partially supported by the DGES, PB97-1069-C02-02. Esther García has been partially supported by a F.P.I. Grant (Ministerio de Ciencia y Tecnología).

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