Simplicity of the Heart of a Nondegenerate Jordan System ^{1}

José A. Anquela, Teresa Cortés, Esther García ^{2}

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

Dedicated to Prof. Kevin McCrimmon on the occasion of his 60th birthday

Abstract: In this paper we prove that the heart of a nondegenerate Jordan system (algebra, triple system or pair) is either simple or zero. We also obtain Herstein type results relating the hearts of associative systems and those of their corresponding Jordan systems.

2000 Math. Subj. Class.: 17C05, 17C10, 17C20

Introduction

The notion of heart (i.e., the intersection of all nonzero ideals) appears naturally when studying associative or Jordan systems. Indeed, having a nonzero heart can be considered as a strong version of primeness for semiprime or nondegenerate systems. As an example, in the proof of the Jordan algebra version of Kaplansky’s Theorem [10, 1.2], having or not a nonzero heart plays a major role. In the recent results on local and subquotient inheritance of simplicity [6], having a “sufficiently big” heart comes out as the natural (weaker) substitute of simplicity. The heart is used in a similar way in the Herstein type results of [7] relating Jordan and associative ideals. There is a common feature in all of these different particular situations: under suitable conditions, the heart of a nondegenerate Jordan (resp. semiprime associative) system is shown to be simple when it is nonzero.

Indeed, using Andrunakievich’s Lemma, it is not hard to prove that the heart of an arbitrary semiprime associative system is either zero or simple. However, the Jordan version of Andrunakievich’s Lemma is false even for linear Jordan algebras (cf. [18]) and thus the study of the simplicity of the heart in Jordan systems requires different techniques. For nondegenerate linear Jordan algebras the simplicity of the heart when it is nonzero is a consequence of the following stronger result by Medvedev [18, p. 933] and Skosyrskii [21, Cor. 3.1]: every minimal ideal of a linear Jordan algebra is either simple or trivial. The techniques are mainly combinatorial and strongly dependent on the linearity, i.e., on the existence of 1/2 in the ring of scalars. Prior to the results by Medvedev and Skosyrskii, Ng Seong Nam and Kevin McCrimmon [20] give a first approach to the study of minimal ideals in quadratic Jordan algebras. In this more general setting, they show that minimal ideals in arbitrary (quadratic) Jordan algebras are either trivial or D-simple, i.e., not having proper ideals invariant by all derivations. As a consequence, the heart of an arbitrary Jordan algebra is either trivial or D-simple.

1 Partially supported by the DGES, PB97-1069-C02-02.

2 Partially supported by a F.P.I. Grant (Ministerio de Educación y Cultura).

In this paper we extend the above mentioned results on the simplicity of the heart to arbitrary nondegenerate Jordan systems (algebras, triple systems and pairs) without any restriction on the ring of scalars, though we do not attack the more general problem of the simplicity of minimal ideals. We will base our proofs on the structure theory of strongly prime Jordan systems [2, 3, 17, 22], the use of hermitian or Zelmanov polynomials [1, 17] and the recent Herstein type theorems [7] relating ideals of associative systems and their corresponding Jordan systems.

After a preliminary section, we study in Section 1 ideals of hermitian polynomials, namely of hearty eaters, both in the algebra and triple system settings. Indeed we just elaborate on the constructions given in [1, 17] to show the existence of ideals of hearty eaters arbitrarily voracious. These are used in the following two sections to obtain the main results in the paper. In the second section we show that the heart of a nondegenerate Jordan algebra is either zero or simple, and in the third section we give triple system and pair analogues of that result.

Many consequences can be obtained from the above results. Some of them are collected in the last section of the paper, where we also obtain Herstein type results relating the heart of an associative system R (resp. the ∗-heart of an associative system with involution ∗) and the heart of its symmetrization R^{(+)} (resp. any ample subspace H_{0}(R, ∗) of ∗-symmetric elements of R).

0. Preliminaries

0.1 We will deal with associative and Jordan algebras, pairs and triple systems over an arbitrary ring of scalars Φ. The reader is referred to [2, 11, 12, 17] for basic results, notation and terminology, though we will stress some definitions and basic notions. The identities JPx listed in [12] will be quoted with their original numbering without explicit reference to [12].

—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 One can obtain Jordan systems from associative systems by symmetrization: If R is an associative algebra, we can 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. Similarly, a Jordan pair (resp. Jordan triple system) R^{(+)} can be obtained from an associative pair R = (R^{+}, R^{−}) (resp. an associative triple system R) by defining Q_{x}y = xyx, for any x ∈ R^{σ}, y ∈ R^{−σ}, σ = ± (resp. P_{x}y = xyx, for any x, y ∈ R).

A Jordan system (algebra, pair or triple system) is said to be special if it is a subsystem of R^{(+)} for some associative system R.

0.3 A particularly important example of special Jordan systems are ample subspaces or subpairs of associative systems with involution:

—If R is an associative algebra with involution ∗, a Φ-submodule H_{0}(R, ∗) 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 + x^{∗}, xx^{∗} ∈ H_{0}(R, ∗) for any x ∈ R) and xH_{0}(R, ∗)x^{∗} ⊆ H_{0}(R, ∗) for any x ∈ R [13, p. 387; 17, 0.8′].

—If R = (R^{+}, R^{−}) is an associative pair with polarized involution ∗, an ample subpair H_{0}(R, ∗) = (H_{0}^{+}, H_{0}^{−}) is a pair of submodules of symmetric elements (H_{0}^{σ} ⊆ H(R^{σ}, ∗)) containing all traces {x} = x + x^{∗} of the elements of R^{σ} and satisfying xH_{0}^{−σ}x^{∗} ⊆ H_{0}^{σ}, for any x ∈ R^{σ}, σ = ±. To obtain the notion of ample subspace of an associative triple system R with involution, simply forget the superscripts in the case of pairs [2, p. 209-210; 8, 1.7].

Clearly, H_{0}(R, ∗) is a subsystem of R^{(+)} in all three cases.

When dealing with pairs, involutions will be assumed to be of polarized type.

0.4 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. [15, 3.5]).

0.5 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^{+} [12, 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.

0.6 Regularity conditions through the functor T ( ) [2, p. 230]: Let V be a Jordan (resp. associative) pair.

(i) V is nondegenerate (resp. semiprime) if and only if T (V) is nondegenerate (resp. semiprime),

(ii) V is strongly prime (resp. prime) if and only if T (V) is strongly prime (resp. prime),

(iii) V is simple if and only if T (V) is simple.

If R is an associative system with involution ∗, primeness can be replaced by ∗-primeness in (ii) [5, 1.4(i)], and simplicity can be replaced by ∗-simplicity in (iii).

0.7 Homotopes 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}.

—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}.

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

The above notions are compatible with the functors V ( ) and T ( ) (cf. [5, 0.5]).

0.8 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.

0.9 Assume that J is a strongly prime Jordan system and L is an ideal of J such that L is simple. Then, for any nonzero ideal I of J, I ∩ L ≠ 0 by strong primeness and it is an ideal of L. Hence L = I ∩L by simplicity of L and L ⊆ I. This argument applies with suitable changes to associative systems with and without involution, so that we have:

(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).

0.10 For a Jordan algebra J, the centroid Γ(J) of J is defined as [14, p. 298]

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

By [14, Th. 2], for a Jordan algebra J with zero extreme radical (for example, if J is semiprime), Γ(J) is an associative commutative ring and J can be considered as a Jordan algebra over Γ(J). When J is prime, Γ(J) is a domain acting on J without torsion [17, 4.2; 16, 2.8] and one can consider the so-called central closure Γ(J)^{−1}J of J, which is an algebra over the field of fractions Γ(J)^{−1}Γ(J).

Indeed one has a Jordan version of Posner-Rowen theorem for associative algebras:

0.11 Theorem [10, 1.1]. Let J be a strongly prime, PI Jordan algebra. Then Γ(J)^{−1}J is simple and unital.

A triple system version of the above results is still not known (see [19, 6.1, 6.2]). However, a weak version of it is obtained in [19] by means of the following substitutes of the usual centroid and the usual central closure.

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

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

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

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

The set of all J-homomorphisms defined on I is denoted by Hom_{J}(I, J) [19, 1.1].

0.13 A pair (f, I), where f ∈ Hom_{J}(I, J) is said to be permissible if the ideal I of J is essential (it hits every nonzero ideal of J) [19, 1.1].

0.14 In the set of all permissible maps (f, I) of a Jordan triple systems J the relation “∼” is defined by

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

and turns out to be an equivalence relation. The quotient set C(J) is called the extended centroid of J. The equivalence class of a permissible map (f, I) will be denoted by [f, I] [19, 1.4].

For a nondegenerate Jordan triple system J, there is a natural way to define a Φ-algebra structure on C(J) [19, 1.11, 1.13]. Indeed:

(i) If J is nondegenerate, then C(J) is a commutative, associative, unital, von Neumann regular Φ-algebra [19, 1.15].

(ii) If J is strongly prime, then C(J) is a field [19, 1.15].

0.15 Let J be a nondegenerate Jordan triple system. In the (free) scalar extension C(J) ⊗_{Φ} J the set

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

is an ideal of C(J) ⊗_{Φ} J [19, 3.2]. In [19, 3.6], an ideal M̃ is found in the quotient J̃ = (C(J) ⊗_{Φ} J)/R so that, up to isomorphism, J is still contained in J̃/M̃ and this is tight over J. This latter quotient J̃/M̃, denoted C(J)J, is called the extended central closure of J. Rather than the explicit construction of M̃ and C(J), we will need the following properties of the central closure:

(i) C(J)J is a Jordan triple system over C(J) which is a tight extension of J. Thus, we can assume that J ⊆ C(J)J, rather than explicitly using the injective map x ↦ ((1 ⊗ x) + R) + M̃ [19, 3.5, 3.6].

(ii) If x ∈ I and λ = [f, I] ∈ C(J) the product λx in C(J)J is given by λx = f(x) (see the definition of R).

When dealing with Jordan triple systems we will use the following weak version of Posner-Rowen Theorem:

0.16 Theorem [19, 6.1]. Let J be a strongly prime, homotope-PI Jordan triple system. Then C(J)J is simple.

1. Hearty-Eaters in Jordan Systems

In [1] and [17] nonzero ideals of hearty pentad eaters for Jordan triple systems and algebras are found. By taking suitable powers of those ideals, we will find nonzero ideals of hearty n-tad eaters for all n:

1.1 Recall the notion of adic family on a special Jordan algebra [17, 13.4–6]. The subset of all hearty n-tad eaters [17, 13.8] in the free special Jordan algebra FSJ_{alg}(X) (X is an infinite set of variables) will be denoted HE_{n}(X). A hearty n-tad eater is a Jordan polynomial p = p(x_{1}, …, x_{r}) which eats adic families in FSJ_{alg}(X):

(i) F_{n}(y_{1}, …, y_{n−1}, p) = ∑ F_{3}(q_{1}, q_{2}, q_{3}),

where q_{i} = q_{i}(x_{1}, …, x_{r}, y_{1}, …, y_{n−1}), i = 1, 2, 3, lie in FSJ_{alg}(X). Replacing variables by the unit element shows that a hearty n-tad eater p eats in lower levels of adic families:

(ii) If m ≤ n, then F_{m}(y_{1}, …, y_{m−1}, p) = ∑ F_{3}(q_{1}, q_{2}, q_{3}).

Following [17, 13.8], we denote by H_{n}(X) the core of HE_{n}(X), i.e., the biggest ideal of FSJ_{alg}(X) consisting of hearty n-tad eaters. Using (ii), we obtain

(iii) HE_{n+1}(X) ⊆ HE_{n}(X), hence H_{n+1}(X) ⊆ H_{n}(X) for all n.

Notice that, by arguing as in [1, 3.14(3)], but using the fact that H_{n}(X) is an ideal, the elements in H_{n}(X) eat adic m-tads from any position:

(iv) If p ∈ H_{n}(X) and m ≤ n, then

F_{m}(y_{1}, …, y_{k}, p, y_{k+1}, …, y_{m−1}) = ∑ F_{3}(q_{1}, q_{2}, q_{3}),

for any 0 ≤ k ≤ m − 1.

In [4, 3.6], hearty n-tad eaters are shown to be exactly those polynomials which eat associative n-tads. Indeed, we just need this weaker property of hearty eaters:

(v) If p ∈ H_{n}(X) and m ≤ n, then

y_{1} … y_{k}py_{k+1} … y_{m−1} = ∑ q_{1}q_{2}q_{3},

for any 0 ≤ k ≤ m − 1.

1.2 Similarly, adic families on a special Jordan triple system are defined in [1, 3.7]. Unlike on Jordan algebras, adic families on special Jordan triple systems can only occur at odd levels, and we do not have the possibility of reducing the level by introducing unit elements. Thus, the definition of hearty eaters in triple systems is changed so that the triple version of (1.1)(ii) holds. We will match the algebra notation unlike [1] and denote by HET_{n}(X) (n is odd) the subset of all hearty n-tad eaters [1, 3.12] in the free special Jordan triple system FSJ_{trip}(X) (X is an infinite set of variables). A hearty n-tad eater (n is odd) in the triple sense is a Jordan polynomial p = p(x_{1}, …, x_{r}) which eats adic families in FSJ_{trip}(X) at all odd levels non greater than n:

(i) If m is odd and m ≤ n, then

F_{m}(y_{1}, …, y_{m−1}, p) = ∑ F_{3}(q_{1}, q_{2}, q_{3})

and

F_{m}(y_{1} …, y_{m−2}, p, y_{m−1}) = ∑ F_{3}(q_{4}, q_{5}, q_{6}),

where q_{i} = q_{i}(x_{1}, …, x_{r}, y_{1}, …, y_{n−1}), i = 1, …, 6, lie in FSJ_{trip}(X).

We denote by HT_{n}(X) the core of HET_{n}(X), i.e., the biggest ideal of FSJ_{trip}(X) consisting of hearty n-tad eaters. Clearly,

(ii) HET_{n+2}(X) ⊆ HET_{n}(X), hence HT_{n+2}(X) ⊆ HT_{n}(X) for all odd n.

Arguing as in the algebra case, we have that the elements in HT_{n}(X) eat adic m-tads from any position:

(iii) If p ∈ HT_{n}(X), m is odd and m ≤ n, then

F_{m}(y_{1}, …, y_{k}, p, y_{k+1}, …, y_{m−1}) = ∑ F_{3}(q_{1}, q_{2}, q_{3}),

for any 0 ≤ k ≤ m − 1.

As for algebras we just need the weaker associative n-tad eating property of hearty eaters:

(iv) If p ∈ HT_{n}(X), m is odd and m ≤ n, then

y_{1} … y_{k}py_{k+1} … y_{m−1} = ∑ q_{1}q_{2}q_{3},

for any 0 ≤ k ≤ m − 1.

We will show that nonzero hearty n-tad eaters exist in the core H_{n}(X) of HE_{n}(X) (resp. HT_{n}(X) of HET_{n}(X)) for arbitrarily big n.

1.3 Proposition. For all n ≥ 5, H_{n}(X) is a nonzero linearization invariant ideal of FSJ_{alg}(X). Moreover, H_{n}(X) contains nonzero Clifford polynomial identities.

Proof: In [17, 13.10] it is shown that H_{n}(X) is linearization invariant for any n ≥ 4.

We claim that, for all n ≥ 5,

U_{H_{n}(X)}H_{n}(X) ⊆ H_{n+2}(X)  (1)

To show (1), since U_{H_{n}(X)}H_{n}(X) is an ideal of FSJ_{alg}(X), we just need to show that U_{H_{n}(X)}H_{n}(X) ⊆ HE_{n+2}(X) : Let p, q ∈ H_{n}(X); for any adic family {F_{k}}_{k∈N},

F_{n+2}(y_{1}, …, y_{n+1}, U_{p}q) = F_{n+4}(y_{1}, …, y_{n+1}, p, q, p) (by [17, 13.4 (AIII)]) =

∑ F_{n+2}(y_{1}, …, y_{n−3}, q_{1}, q_{2}, q_{3}, q, p) (since p ∈ H_{n}(X) ⊆ H_{5}(X) by (1.1)(iii)) =

∑ F_{n}(y_{1}, …, y_{n−4}, q_{4}, q_{5}, q_{6}, p) (since q ∈ H_{n}(X) ⊆ H_{5}(X) by (1.1)(iii)) =

∑ F_{3}(q_{7}, q_{8}, q_{9})

since p ∈ H_{n}(X).

On the other hand, in [17, 14.2; 1, p. 182], an element p ∈ H_{5}(X) is found which takes the value e_{12} + e_{21} in the Jordan algebra of symmetric 3 × 3 matrices H_{3}(Φ) over Φ, under a suitable substitution of the variables. This readily implies that H_{5}(H_{3}(Φ)) = H_{3}(Φ), hence H_{n}(H_{3}(Φ)) = H_{3}(Φ) for all n ≥ 5, using (1), (1.1)(iii) and the fact that U_{H_{3}(Φ)}H_{3}(Φ) = H_{3}(Φ). In particular, H_{n}(X) contains nonzero Clifford polynomial identities, for all n ≥ 5

1.4 Proposition. For all odd n ≥ 5, HT_{n}(X) is a nonzero linearization invariant ideal of FSJ_{trip}(X). Moreover, HT_{n}(X) contains nonzero Clifford homotope polynomial identities.

Proof: By [1, 3.14(1)] for all odd n ≥ 5, HT_{n}(X) is linearization invariant. Moreover, HT_{5}(X) = HET_{5}(X) by [1, 3.16].

In [5, 0.8] it is shown that, for any odd n, HET_{n+4}(X) contains the semiideal HT_{5}(X)^{n}, where the powers HT_{5}(X)^{n} are defined inductively by

HT_{5}(X)^{1} = HT_{5}(X), HT_{5}(X)^{n} = P_{HT_{5}(X)}(HT_{5}(X)^{n−2}).

Notice that:

(1) the elements in HT_{5}(X)^{n} eat adic m-tads for any odd m ≤ n + 4 from any position [5, 0.7].

Let I_{n}(X) be the ideal of FSJ_{trip}(X) generated by HT_{5}(X)^{n}, i.e.,

I_{n}(X) = HT_{5}(X)^{n} + P_{FSJ_{trip}(X)}(HT_{5}(X)^{n}).

We claim that I_{n}(X) ⊆ HET_{n+2}(X), hence I_{n}(X) ⊆ HT_{n+2}(X): HT_{5}(X)^{n} ⊆ HET_{n+4}(X) ⊆ HET_{n+2}(X) by (1.2)(ii), hence one just need to prove that an element P_{q}p, where p ∈ HT_{5}(X)^{n} and q ∈ FSJ_{trip}(X), eats adic m-tads for any odd m ≤ n+2; but indeed P_{q}p eats adic m-tads for any odd m ≤ n + 2 from any position, by using [1, 3.6(A3)], since p eats adic m-tads for any odd m ≤ n + 4 from any position, by (1).

By [1, 4.5; 3, Section 6], there exists an element p ∈ HT_{5}(X) which is a homotope polynomial, i.e., there exists q = q(x_{1}, …, x_{r}) ∈ FSJ_{alg}(X) such that p = p(x_{1}, …, x_{r}, y) = q(y; x_{1}, …, x_{r}) is the evaluation of q on the homotope algebra FSJ_{trip}(X)^{(y)} of FSJ_{trip}(X). Moreover, a suitable evaluation of q (i.e., of p, where y takes the value 1 = e_{11} + e_{22} + e_{33}) on H_{3}(Φ) yields e_{23} + e_{32}. This shows that p is a nonzero Clifford homotope polynomial identity inside HT_{5}(X). Now, for any positive integer n, let q_{n} = q^{n}, p_{n}(x_{1}, …, x_{r}, y) = q_{n}(y; x_{1}, …, x_{r}). Clearly

p_{4n+1}(x_{1}, …, x_{r}, y) = (U_{q(y;x_{1},...,x_{r})}^{(y)})^{2n}q(y; x_{1}, …, x_{r}) =
(P_{q(y;x_{1},...,x_{r})}P_{y})^{2n}q(y; x_{1}, …, x_{r}) = (P_{p}P_{y})^{2n}p = (P_{p}P_{y}P_{p}P_{y})^{n}p =
(P_{p}P_{P_{y}p})^{n}p ∈ HT_{5}(X)^{4n+1}

since p, P_{y}p ∈ HT_{5}(X). Thus p_{4n+1}(x_{1}, …, x_{r}, y) ∈ I_{4n+1}(X) ⊆ HT_{4n+3}(X) ⊆ HT_{n}(X) by (1.2)(ii) for all odd n ≥ 5. Notice that p_{4n+1} is a Clifford homotope identity since the same replacement of variables of q_{4n+1} in H_{3}(Φ) as the one mentioned above for q yields (e_{23} + e_{32})^{4n+1} = e_{23} + e_{32} ≠ 0.

2. The Heart of a Nondegenerate Jordan Algebra

2.1 It is well known the fact that the heart (resp. the ∗-heart) of a semiprime associative algebra (resp. a semiprime associative algebra with involution ∗) R is simple (resp. ∗-simple) when it is nonzero: If 0 ≠ I is the heart (resp. the ∗-heart) of R, then it is not trivial since R is semiprime. Moreover, for any nonzero ideal (resp. ∗-ideal) L of I the ideal L̃ of R generated by L (which is a ∗-ideal in the case with involution) satisfies L̃^{3} ⊆ L (Andrunakievich’s Lemma). Since 0 ≠ L ⊆ L̃, L̃^{3} ≠ 0 by semiprimeness of R and I ⊆ L̃^{3} ⊆ L, i.e., I = L, which shows that I is simple (resp. ∗-simple). Our main task in this section will be to prove an analogue for Jordan algebras of the above result.

We begin with a remark dealing with the cube of the heart of a nondegenerate Jordan algebra.

2.2 Remark: For every nondegenerate Jordan algebra J,

Heart(J)^{3} := U_{Heart(J)} Heart(J) = Heart(J).

Indeed, we can assume that I = Heart(J) is nonzero since, otherwise, our assertion is obvious. Notice that I^{3} = U_{I}I is an ideal of J which is nonzero since I is nondegenerate by (0.4). Hence I ⊆ I^{3} and I^{3} is clearly contained in I, which yields I = I^{3}.

The next lemma will allow us to use the structure theory of strongly prime Jordan algebras when studying the heart.

2.3 Lemma. If J is a nondegenerate Jordan algebra and Heart(J) ≠ 0 then J is prime (hence strongly prime).

Proof: Let I, L be nonzero ideals of J. We have that U_{Heart(J)} Heart(J) ⊆ U_{I}L and U_{Heart(J)} Heart(J) ≠ 0 since Heart(J) is nondegenerate by (0.4). Thus U_{I}L ≠ 0.

2.4 Remark: The proof of (2.3) can be easily adapted to the associative case to prove the following assertion: If R is a semiprime associative algebra (resp. a semiprime associative algebra with involution ∗) and Heart(R) ≠ 0 (resp. ∗- Heart(R) ≠ 0), then R is prime (resp. ∗-prime).

When a nondegenerate algebra is PI, having a nonzero heart has further consequences than the mere simplicity of the heart. This is shown in the next result, which can be considered as a part of our main theorem for algebras.

2.5 Proposition. If J is a nondegenerate, PI Jordan algebra and the heart Heart(J) of J is nonzero, then Heart(J) = J and J is simple and unital.

Proof: Let I := Heart(J) ≠ 0.

By (2.3), J is strongly prime. Since J is PI, then, by (0.11), the central closure Γ(J)^{−1}J of J is simple and unital.

Let 0 ≠ γ ∈ Γ(J). We claim that γI ⊆ I: for any x, y ∈ I, γ(U_{x}y) = U_{x}γy ∈ U_{I}J ⊆ I, which shows γI^{3} ⊆ I; but I^{3} = I by (2.2). Moreover, γI is an ideal of J: for any x ∈ I, y ∈ J,

U_{γx}y = γ^{2}U_{x}y = γ(U_{x}(γy)) ∈ γ(U_{I}J) ⊆ γI,
(γx)^{2} = γ^{2}x^{2} = γ(γx^{2}) ∈ γ(γI) ⊆ γI,
(γx) ∘ y = γ(x ∘ y) ∈ γ(I ∘ J) ⊆ γI,
U_{y}(γx) = γ(U_{y}x) ∈ γ(U_{J}I) ⊆ γI.

Using that Γ(J) acts on J without torsion, γI is nonzero, which implies I ⊆ γI, hence I = γI.

Since we have Γ(J)I = I, Γ(J)^{−1}I makes sense and is an ideal of Γ(J)^{−1}J. Moreover, 0 ≠ I ⊆ Γ(J)^{−1}I, hence Γ(J)^{−1}I = Γ(J)^{−1}J by simplicity of Γ(J)^{−1}J.

We claim that Γ(J)^{−1}I = I. Indeed, for any 0 ≠ γ ∈ Γ(J) and x ∈ I, since γI = I, there exists z ∈ I such that x = γz. But this implies γ^{−1}x = z ∈ I.

Thus Γ(J)^{−1}J = Γ(J)^{−1}I = I ⊆ J ⊆ Γ(J)^{−1}J, which implies I = J = Γ(J)^{−1}J which is simple and unital.

Now, we can prove our main result for Jordan algebras.

2.6 Theorem. If J is a nondegenerate Jordan algebra then the heart Heart(J) of J is either simple or zero.

Proof: Let us assume that I := Heart(J) ≠ 0.

If J is PI, then the theorem follows from (2.5). Thus we may assume that J is not PI.

By (2.3), J is strongly prime hence J is special by [17, 15.2] since it is not PI. Thus we can take an associative ∗-tight envelope R of J, i.e., J ⊆ H(R, ∗), R is generated as an associative algebra by J and every nonzero ∗-ideal of R hits J.

Let Ĩ be the ideal of R generated by I. Notice that Ĩ is a nonzero ∗-ideal of R.

(I) Ĩ = ∗-Heart(R): For any nonzero ∗-ideal L of R, L ∩ J is a nonzero ideal of J by ∗-tightness, hence I ⊆ L ∩ J ⊆ L and Ĩ ⊆ L. Thus Ĩ ⊆ ∗-Heart(R), but ∗-Heart(R) ⊆ Ĩ since Ĩ is a nonzero ∗-ideal of R.

(II) Ĩ is ∗-simple by (2.1) since R is semiprime by ∗-tightness.

(III) IJ + JI ⊆ II: Let a ∈ I, x ∈ J. By (2.2), a = ∑ U_{a_{i}}b_{i}, where a_{i}, b_{i} ∈ I. Hence

ax = (∑ U_{a_{i}}b_{i})x = (∑ a_{i}b_{i}a_{i})x = ∑(a_{i}{b_{i}, a_{i}, x} − a_{i}xa_{i}b_{i}) =
∑(a_{i}{b_{i}, a_{i}, x} − (U_{a_{i}}x)b_{i}) ∈ I{I, I, J} + (U_{I}J)I ⊆ II.

Similarly, xa ∈ II.

(IV) Ĩ = I + II + III + …: Indeed, Ĩ = I + RI + IR + RIR ⊆ I + II + III + … by (III), since R is generated by J.

(V) I ⊆ H_{n}(J) for any n ≥ 4: Since J is not PI, by (1.3) and (1.1)(iii), H_{n}(J) is a nonzero ideal of J, hence I ⊆ H_{n}(J).

(VI) I is an ample subspace of symmetric elements of Ĩ, so that we can write I = H_{0}(Ĩ, ∗): Since clearly I ⊆ H(Ĩ, ∗), we only have to prove that I contains all traces and norms of elements in Ĩ and also xIx^{∗} for any x ∈ Ĩ:

— For any x_{1}, …, x_{n} ∈ I, n ≥ 4,

x_{1} … x_{n} + (x_{1} … x_{n})^{∗} = \{x_{1} … x_{n}\} ∈ \{H_{n}(J)I \overbrace{J … J}^{n−2}\} (by (V)) ⊆

∑{q_{1}(I, J, …, J), q_{2}(I, J, …, J), q_{3}(I, J, …, J)} ⊆
{I, J, J} + {J, I, J} + {J, J, I} ⊆ I

since by homogeneity in (1.1)(v), in every summand, at least one the factors q_{1}, q_{2}, q_{3} contains a variable evaluated in I which appears in all of its Jordan monomials. The above, with (IV) implies that I contains all traces of elements in Ĩ.

— Using (IV), every norm of an element in Ĩ is a sum of traces of elements in Ĩ and elements of the form x_{1} … x_{n}x_{n} … x_{1}, where x_{1}, …, x_{n} ∈ I. Since x_{1} … x_{n}x_{n} … x_{1} = U_{x_{1}} … U_{x_{n−1}}x_{n}^{2} ∈ U_{I} … U_{I}I^{2} ⊆ I, every norm of an element in Ĩ lies in I.

— By (IV) again, if x ∈ Ĩ, the elements in xIx^{∗} are sums of traces of elements in Ĩ and elements of the form x_{1} … x_{n}yx_{n} … x_{1}, where x_{1}, …, x_{n}, y ∈ I. Since x_{1} … x_{n}yx_{n} … x_{1} = U_{x_{1}} … U_{x_{n}}y ∈ U_{I} … U_{I}I ⊆ I, we have xIx^{∗} ⊆ I for any x ∈ Ĩ.

(VII) Finally, I is simple by [7, 2.7 (ii)], using (II) and the fact that I^{3} = I by (2.2).

3. The Heart of a Nondegenerate Jordan Triple System or Pair

We will obtain triple system versions of the results in the previous section.

3.1 As in the algebra case, the heart (resp. the ∗-heart) of a semiprime associative triple system (resp. a semiprime associative triple system with involution ∗) R is simple (resp. ∗-simple) when it is nonzero. The proof [6, 4.5] follows by using an Andrunakievich-like argument which is obviously still valid in the case with involution.

The proof of (2.2) cannot be applied to triple systems since the cubes of ideals are not ideals anymore in the triple case (cf. [2, 2.9]). However, the triple version of (2.2) is still true.

3.2 Remark [6, 4.4]: For every semiprime Jordan triple system J,

Heart(J)^{3} := P_{Heart(J)} Heart(J) = Heart(J).

The proof of (2.3) can be extended to triple systems simply by replacing U’s by P ’s.

3.3 Lemma. If J is a nondegenerate Jordan triple system and Heart(J) ≠ 0 then J is prime (hence strongly prime).

3.4 Remark: Similarly, if R is a semiprime associative triple system (resp. a semiprime associative triple system with involution ∗) and Heart(R) ≠ 0 (resp. ∗-Heart(R) ≠ 0), then R is prime (resp. ∗-prime).

To obtain the triple system version of (2.5), we will need to use extended central closures instead of usual central closures.

3.5 Proposition. If J is a nondegenerate, homotope-PI Jordan triple system and the heart Heart(J) of J is nonzero, then Heart(J) = J and J is simple.

Proof: Let I := Heart(J) ≠ 0.

By (3.3), J is strongly prime. Since J is homotope-PI, the extended central closure C(J)J of J is simple by (0.16).

Let 0 ≠ γ ∈ C(J). We claim that γI ⊆ I: γ = [f, L], where L is a nonzero ideal of J; hence I ⊆ L and γx = f(x) for any x ∈ I by (0.15)(ii). Moreover, since I = I^{3} by (3.2), x = ∑ P_{a_{i}}b_{i}, hence f(x) = ∑ P_{a_{i}}f(b_{i}) ∈ P_{I}J ⊆ I, using (0.12)(i).

Now, we have that I is a nonzero ideal of C(J)J, hence I = C(J)J by simplicity of C(J)J.

Thus J ⊆ C(J)J = I ⊆ J, and J = I is a simple Jordan triple system.

Now, we can prove our main result for Jordan triple systems, which is just the triple system version of (2.6).

3.6 Theorem. If J is a nondegenerate Jordan triple system then the heart Heart(J) of J is either simple or zero.

Proof: Let us assume that I := Heart(J) ≠ 0.

If J is homotope-PI, then the theorem follows from (3.5). Thus we may assume that J is not homotope-PI.

By (3.3), J is strongly prime and hence J is special by [2, 4.1; 22, Th. 5] since J is not homotope-PI. Thus we can take an associative ∗-tight envelope R of J, i.e., J ⊆ H(R, ∗), R is generated as an associative triple system by J and every nonzero ∗-ideal of R hits J.

Let Ĩ be the ideal of R generated by I. Notice that Ĩ is a nonzero ∗-ideal of R. (I) Ĩ = ∗-Heart(R): Formally, the proof of part (I) of the proof of (2.6) holds. (II) Ĩ is ∗-simple by (3.1) since R is semiprime by ∗-tightness. (III) IJJ + JJI + JIJ ⊆ III: Let a ∈ I, x, y ∈ J. By (3.2), a = ∑ P_{a_{i}}b_{i}, where

a_{i}, b_{i} ∈ I. Hence

axy = (∑ P_{a_{i}}b_{i})xy = (∑ a_{i}b_{i}a_{i})xy = ∑(a_{i}{b_{i}, a_{i}, x}y − a_{i}xa_{i}b_{i}y) =
∑(a_{i}{b_{i}, a_{i}, x}y − (P_{a_{i}}x)b_{i}y) ∈ I{I, I, J}y + (P_{I}J)Iy ⊆ IIy.

Also,

xay = x(∑ P_{a_{i}}b_{i})y = (∑ xa_{i}b_{i}a_{i})y = ∑({x, a_{i}, b_{i}}a_{i}y − b_{i}a_{i}xa_{i}y) =
∑({x, a_{i}, b_{i}}a_{i}y − b_{i}(P_{a_{i}}x)y) ∈ {J, I, I}Iy + I(P_{I}J)y ⊆ IIy.

We have shown, for any y ∈ J,

IJy + JIy ⊆ IIy.  (1)

Similarly,

yJI + yIJ ⊆ yII.  (2)

Now,

IJJ ⊆ IIJ (by (1)) ⊆ III (by (2)),
JJI ⊆ JII (by (2)) ⊆ III (by (1)),
JIJ ⊆ IIJ (by (1)) ⊆ III (by (2)).

(IV) Ĩ = I + III + IIIII + …: Indeed we just need to notice that I + III + IIIII + … is an ideal of R, which follows from (III), since R is generated by J.

(V) I ⊆ HT_{n}(J) for any odd n ≥ 5: Since J is not homotope-PI, HT_{n}(J) is a nonzero ideal of J by (1.4), hence I ⊆ HT_{n}(J).

(VI) I is an ample subspace of symmetric elements of Ĩ, so that we can write I = H_{0}(Ĩ, ∗): Since clearly I ⊆ H(Ĩ, ∗), we only have to prove that I contains all traces of elements in Ĩ and also xIx^{∗} for any x ∈ Ĩ:

— For any x_{1}, …, x_{n} ∈ I, where n is odd and n ≥ 5,

x_{1} … x_{n} + (x_{1} … x_{n})^{∗} = \{x_{1} … x_{n}\} ∈ \{HT_{n}(J)I \overbrace{J … J}^{n−2}\} (by (V)) ⊆

∑{q_{1}(I, J, …, J), q_{2}(I, J, …, J), q_{3}(I, J, …, J)} ⊆
{I, J, J} + {J, I, J} + {J, J, I} ⊆ I

since by homogeneity in (1.2)(iv), in every summand, at least one the factors q_{1}, q_{2}, q_{3} contains a variable evaluated in I which appears in all of its Jordan monomials. The above, with (IV) implies that I contains all traces of elements in Ĩ.

— By (IV) again, if x ∈ Ĩ, the elements in xIx^{∗} are sums of traces of elements in Ĩ and elements of the form x_{1} … x_{n}yx_{n} … x_{1}, where x_{1}, …, x_{n}, y ∈ I. Since x_{1} … x_{n}yx_{n} … x_{1} = P_{x_{1}} … P_{x_{n}}y ∈ P_{I} … P_{I}I ⊆ I, we have xIx^{∗} ⊆ I for any x ∈ Ĩ.

(VII) Finally, I is simple by [7, 3.15 (ii)], using (II) and the fact that I^{3} = I by (3.2).

We will obtain pair analogues of the above results for triple systems by using the functor T ( ).

3.7 Let V be a Jordan (resp. associative) pair and T (V) be the polarized triple system associated to V. Using the definition of the functor T ( ) immediately yields

(i) T (V)^{3} = T (V^{3}), where V^{3} = (Q_{V^{+}}V^{−}, Q_{V^{−}}V^{+}).

On the other hand, in [2, p. 230] it is shown that, for any semiprime Jordan pair (in particular, for any nondegenerate Jordan pair) V, every nonzero ideal of T (V) contains a polarized ideal T (I), where I is a nonzero ideal of V. The proof can be easily adapted to semiprime associative pairs with and without involution. This readily implies

(ii) If V is a semiprime Jordan or associative pair, then Heart(T (V)) = T (Heart(V)). If V is a semiprime associative pair with involution ∗, then ∗-Heart(T (V)) = T (∗-Heart(V)).

Every homotope V^{σ(a)}, where a ∈ V^{−σ}, is clearly a subalgebra of the homotope T (V)^{(a)} of T (V). Conversely, the homotope T (V)^{(a+b)}, where a ∈ V^{+}, b ∈ V^{−} is readily seen to be the direct sum V^{+(b)} + V^{−(a)} of the homotope algebras V^{+(b)}, V^{−(a)} of V. Thus

(iii) V is homotope-PI if and only if T (V) is homotope-PI.

The above comments, together with (0.6)(i)(iii), readily imply the pair versions of (3.1), (3.2), (3.3), (3.4), (3.5) and (3.6).

3.8 The heart (resp. the ∗-heart) of a semiprime associative pair (resp. a semiprime associative pair with involution ∗) R is simple (resp. ∗-simple) when it is nonzero.

3.9 For every nondegenerate Jordan pair V,

Heart(V)^{3} = Heart(V).

3.10 Lemma. If V is a nondegenerate Jordan pair and Heart(V) ≠ 0 then V is prime (hence strongly prime).

3.11 Remark: Similarly, if R is a semiprime associative pair (resp. a semiprime associative pair with involution ∗) and Heart(R) ≠ 0 (resp. ∗-Heart(R) ≠ 0), then R is prime (resp. ∗-prime).

3.12 Proposition. If V is a nondegenerate, homotope-PI Jordan pair and the heart Heart(V) of V is nonzero, then Heart(V) = V and V is simple.

3.13 Theorem. If V is a nondegenerate Jordan pair then the heart Heart(V) of V is either simple or zero.

4. Further Results

The above results on the simplicity of the heart will allow us to relate the heart of a prime system and the heart of any of its subideals avoiding Andrunakievich’s Lemma which is false for arbitrary Jordan systems (cf. [18]).

4.1 Corollary. (i) Let J be a strongly prime Jordan (resp. prime associative) algebra, triple system or pair, and L be a nonzero subideal of J. Then Heart(J) = Heart(L).

(ii) Let R be a ∗-prime associative algebra, triple system or pair with involution ∗ and L be a nonzero ∗-subideal of R. Then ∗-Heart(R) = ∗-Heart(L).

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

Let J be a strongly prime Jordan system, L be a nonzero ideal of J and I be the heart of J and M be the heart of L. Notice that L is strongly prime by [15, 2.5].

For any nonzero ideal N of J, N ∩ L is a nonzero ideal of L by strong primeness of J. Hence M ⊆ N ∩ L, hence

M ⊆ I.  (1)

If I = 0 , then M = 0 by (1). If I ≠ 0, then I is simple by (2.6), (3.6) or (3.13). Moreover, I is contained in L, which implies that I is a simple ideal of L. Since L is strongly prime, I = M by (0.9).

We have shown

Heart(J) = Heart(L)  (2)

for any nonzero ideal L of J.

Since L is strongly prime, (2) applies to L to show that the heart of L coincide with the heart of any nonzero ideal of L. The argument can be iterated to obtain (2) for any subideal of J.

The simplicity of the heart can also be used to study the heart through the functors (0.5) relating algebras, triple systems and pairs.

4.2 Corollary. (i) Let J be a nondegenerate Jordan (resp. semiprime associative) algebra. Then the heart Heart_{alg}(J) of J is equal to the heart Heart_{trip}(J) of the underlying triple system of J.

(ii) Let R be an associative algebra with involution ∗. Then the ∗-heart ∗- Heart_{alg}(R) of R is equal to the ∗-heart ∗-Heart_{trip}(R) of the underlying triple system of R.

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

Let J be a nondegenerate Jordan algebra. Since every nonzero ideal of the algebra J is a nonzero ideal in the triple sense, we have

Heart_{trip}(J) ⊆ Heart_{alg}(J).  (1)

If Heart_{alg}(J) = 0, then Heart_{trip}(J) = 0 by (1).

If Heart_{alg}(J) ≠ 0, then it is simple by (2.6) and J is a strongly prime as an algebra by (2.3). We have that J is strongly prime as a triple system by [8, 1.13, 2.9] and Heart_{alg}(J) is an ideal of J in the triple sense which is simple as a triple system by [6, 3.4]. By (0.9)(i), Heart_{alg}(J) = Heart_{trip}(J).

4.3 Remark: The nice behaviour of the functor T ( ) with respect to the heart (3.7)(ii) is not shared by the functor V ( ), which does not preserve the heart as shown by the following example: Let W be a simple Jordan pair and J = T (W) = W^{+}⊕W^{−}. By (0.6)(iii), J is simple, hence J = Heart(J), but V (J) is a direct sum of two ideals,

V (J) = (W^{+} ⊕ 0, 0 ⊕ W^{−}) + (0 ⊕ W^{−}, W^{+} ⊕ 0),

which implies Heart(V (J)) = 0.

However, when applied to algebras, the functor V preserves the heart.

4.4 Corollary. (i) If J is a nondegenerate Jordan (resp. semiprime associative) algebra, then

Heart(V (J)) = V (Heart(J)).

(ii) If R is a semiprime associative algebra with involution ∗, then

∗-Heart(V (R)) = V (∗-Heart(R)).

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

Let J be a nondegenerate Jordan algebra. Since any nonzero ideal I of J gives rise to a nonzero ideal V (I) of V (J),

Heart(V (J)) ⊆ V (Heart(J)).  (1)

If Heart(J) = 0, then Heart(V (J)) = 0 by (1)

If Heart(J) ≠ 0, then Heart(J) is a simple algebra (2.6), and V (Heart(J)) is a simple Jordan pair by [6, 3.4]. Moreover, V (Heart(J)) is an ideal of V (J), which is strongly prime by [8, 1.12] since J is strongly prime by (2.3). By (0.9)(i), Heart(V (J)) = V (Heart(J)).

4.5 Remark: Nondegeneracy is necessary in (4.2) and (4.4) as the following example shows: Let J be a trivial one-dimensional algebra over a field Φ. Then Heart(J) = J since J is the only nonzero ideal of J. However, V (J) is the direct sum of ideals

V (J) = (J, 0) + (0, J),

which implies that Heart(V (J)) = 0.

By using Herstein’s Theorems [7, 13], we can also study the interaction of the heart with symmetrizations and ample subspaces.

4.6 Corollary. Let R be a semiprime associative algebra, triple system or pair. Then Heart(R^{(+)}) = Heart(R).

Proof: Since every ideal of R is an ideal of R^{(+)}, we have Heart(R^{(+)}) ⊆ Heart(R). By [7, 1.5, 1.6; 13, Corollary of p. 384], every nonzero ideal of R^{(+)} contains a nonzero ideal of R, which implies Heart(R^{(+)}) ⊇ Heart(R).

4.7 Corollary. Let R be an associative algebra, triple system or pair with involution ∗, H_{0}(R, ∗) be an ample subspace of symmetric elements of R, and I = ∗-Heart(R). Then

(i) if R is a semiprime algebra, then Heart(H_{0}(R, ∗)) = K(I, H_{0}(R, ∗)) (cf. [7, 2.1]),

(ii) if R is a ∗-prime triple system, then Heart(H_{0}(R, ∗)) is the ideal of H_{0}(R, ∗) generated by KT (I, H_{0}(R, ∗)) (cf. [7, 3.9]), i.e.

Heart(H_{0}(R, ∗)) = KT (I, H_{0}(R, ∗)) + P_{H_{0}(R,∗)}KT (I, H_{0}(R, ∗),

(iii) if R is a ∗-prime pair, then Heart(H_{0}(R, ∗)) is the ideal of H_{0}(R, ∗) generated by (KP (I^{+}, H_{0}(R, ∗)), KP (I^{−}, H_{0}(R, ∗))) (cf. [7, 3.1]), i.e.

Heart(H_{0}(R, ∗))^{σ} = KP (I^{σ}, H_{0}(R, ∗)) + Q_{H_{0}(R^{σ},∗)}KP (I^{−σ}, H_{0}(R, ∗), σ = ±.

In all cases Heart(H_{0}(R, ∗)) is an ample subspace H_{0}(I, ∗) of I and

Heart(H_{0}(R, ∗)) = 0 ⟺ ∗-Heart(R) = 0.

Proof: (i) Let L be a nonzero ideal of H_{0}(R, ∗). By [7, 2.6], there exists a ∗-ideal L_{1} of R such that 0 ≠ K(L_{1}, H_{0}(R, ∗)) ⊆ L. In particular, L_{1} ≠ 0, hence I ⊆ L_{1} and K(I, H_{0}(R, ∗)) ⊆ L. We have shown that

Heart(H_{0}(R, ∗)) ⊇ K(I, H_{0}(R, ∗)).  (1)

If Heart(H_{0}(R, ∗)) = 0, then K(I, H_{0}(R, ∗)) = 0 by (1), and (i) trivially holds.

Let us assume that Heart(H_{0}(R, ∗)) ≠ 0. Let I be the set of nonzero ∗-ideals of R. By [7, 2.2, 2.5], for any A ∈ I, K(A, H_{0}(R, ∗)) is a nonzero ideal of H_{0}(R, ∗) contained in A. Thus

Heart(H_{0}(R, ∗)) ⊆ ⋂_{A∈I} K(A, H_{0}(R, ∗)) ⊆ ⋂_{A∈I} A = I,

which shows that I ≠ 0. Now K(I, H_{0}(R, ∗)) is a nonzero ideal of H_{0}(R, ∗) by [7, 2.2, 2.5], and (1) readily implies (i).

(ii) Let L be a nonzero ideal of H_{0}(R, ∗). By [7, 3.14], there exists a nonzero ∗- ideal L_{1} of R such that KT (L_{1}, H_{0}(R, ∗)) ⊆ L, hence I ⊆ L_{1} and KT (I, H_{0}(R, ∗) ⊆ L. We have shown that

Heart(H_{0}(R, ∗)) ⊇ KT (I, H_{0}(R, ∗)),

which implies

Heart(H_{0}(R, ∗)) ⊇ KT (I, H_{0}(R, ∗)) + P_{H_{0}(R,∗)}KT (I, H_{0}(R, ∗)).  (2)

If Heart(H_{0}(R, ∗)) = 0, then KT (I, H_{0}(R, ∗)) + P_{H_{0}(R,∗)}KT (I, H_{0}(R, ∗)) = 0 by (1), and (i) trivially holds.

Let us assume that Heart(H_{0}(R, ∗)) ≠ 0. Let I be the set of nonzero ∗-ideals of R. By [7, 3.10, 3.13], for any A ∈ I, KT (A, H_{0}(R, ∗)) is a nonzero semiideal of H_{0}(R, ∗) contained in A. Thus KT (A, H_{0}(R, ∗)) + P_{H_{0}(R,∗)}KT (A, H_{0}(R, ∗)) is a nonzero ideal of H_{0}(R, ∗) contained in A, and

Heart(H_{0}(R, ∗)) ⊆ ⋂_{A∈I}(KT (A, H_{0}(R, ∗)) + P_{H_{0}(R,∗)}KT (A, H_{0}(R, ∗))) ⊆ ⋂_{A∈I} A = I,

which shows that I ≠ 0. Now KT (I, H_{0}(R, ∗))+P_{H_{0}(R,∗)}KT (I, H_{0}(R, ∗) is a nonzero ideal of H_{0}(R, ∗) by [7, 3.10, 3.13], and (2) readily implies (ii).

The proof of (ii) applies to (iii) with obvious changes and the last assertion is an immediate consequence of (i), (ii) and (iii) with [7, 2.5, 3.5, 3.13].

4.8 Remark: ∗-Primeness is needed in (4.7)(ii)(iii) as shown by the following examples already used in [7, 3.4, 3.12]: Let R be the direct sum S + Φ, where S is a ∗-simple associative triple system with involution ∗ such that H(S, ∗) ≠ 0 and Φ is a field of characteristic not two. Extend the involution ∗ to R by (x + λ)^{∗} = x^{∗} − λ, for any x ∈ S, λ ∈ Φ. Clearly R is semiprime and the ∗-heart of R is zero since (S + 0) ∩ (0 + Φ) = 0. But, since the characteristic is not two,

H_{0}(R, ∗) = H(R, ∗) = H(S, ∗) + 0 ≅ H(S, ∗)

is simple by [7, 3.15 (ii)], and Heart(H_{0}(R, ∗)) = H_{0}(R, ∗).

A similar example can be given for pairs by taking R = S + V (Φ), where S is a ∗-simple associative pair with H(S, ∗) ≠ 0 and Φ is again a field of characteristic not two.

4.9 Description of the heart of a Jordan system of Hermitian type: Let J be a strongly prime Jordan algebra, triple or pair of hermitian type, i.e., J has a nonzero ideal of the form H_{0}(R, ∗), where R is a ∗-prime associative system (cf. [2, 4.1, 5.3; 9, 4.1, 4.2, 4.3; 17, 15.2]). Let I be the ∗-heart of R. By (4.1), the heart of J coincides with the heart of H_{0}(R, ∗). Thus, by (4.7),

(i) if J is an algebra, then Heart(J) = K(I, H_{0}(R, ∗)) (cf. [7, 2.1]),

(ii) if J is a triple system, then Heart(J) is the ideal of H_{0}(R, ∗) generated by KT (I, H_{0}(R, ∗)) (cf. [7, 3.9]), i.e.

Heart(J) = KT (I, H_{0}(R, ∗)) + P_{H_{0}(R,∗)}KT (I, H_{0}(R, ∗)),

(iii) if J is a pair, then Heart(J) is the ideal of H_{0}(R, ∗) generated by KP (I, H_{0}(R, ∗)) (cf. [7, 3.1]), i.e.

Heart(J)^{σ} = KP (I^{σ}, H_{0}(R, ∗)) + Q_{H_{0}(R^{σ},∗)}KP (I^{−σ}, H_{0}(R, ∗), σ = ±.

In all cases Heart(J) is an ample subspace H_{0}(I, ∗) of I and

Heart(J) = 0 ⟺ ∗-Heart(R) = 0.

REFERENCES

[1] A. D’Amour, “Zel’manov Polynomials in Quadratic Jordan Triple Systems”. J. Algebra 140 (1991), 160-183.

[2] A. D’Amour, “Quadratic Jordan Systems of Hermitian Type”. J. Algebra 149 (1992), 197-233.

[3] A. D’Amour, K. McCrimmon, “The Structure of Quadratic Jordan Systems of Clifford Type” J. Algebra 234 (2000), 31-89.

[4] J. A. Anquela, M. Cabrera, A. Moreno, “Eater Ideals in Jordan Algebras”. J. Pure Appl. Algebra 125 (1998) 1-17.

[5] J. A. Anquela, T. Cortés, “Primitive Jordan Pairs and Triple Systems”. J. Algebra 184 (1996), 632-678.

[6] J. A. Anquela, T. Cortés, “Local and Subquotient Inheritance of Simplicity in Jordan Systems”, J. Algebra (to appear).

[7] J. A. Anquela, T. Cortés, E. García, “Herstein’s Theorems and Simplicity of Hermitian Jordan Systems” (to appear).

[8] J. A. Anquela, T. Cortés, O. Loos, K. McCrimmon, “An Elemental Characterization of Strong Primeness in Jordan Systems”. J. Pure Appl. Algebra 109 (1996), 23-36.

[9] J. A. Anquela, T. Cortés, K. McCrimmon, F. Montaner, “Strong Primeness of Hermitian Jordan Systems”. J. Algebra 198 (1997) 311-326.

[10] J. A. Anquela, T. Cortés, F. Montaner, “The Structure of Primitive Quadratic Jordan Algebras”. J. Algebra 172 (1995) 530-553.

[11] N. Jacobson, Structure Theory of Jordan Algebras, University of Arkansas Lecture Notes in Mathematics, Vol. 5, Fayetteville, 1981.

[12] O. Loos, Jordan Pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, New York, 1975.

[13] K. McCrimmon, “On Herstein’s Theorems Relating Jordan and Associative Algebras”. J. Algebra 13 (1969), 382-392.

[14] K. McCrimmon, “The Freudenthal-Springer-Tits Construction Revisited”. Trans. Amer. Math. Soc. 148 (1970), 293-314.

[15] K. McCrimmon, “Strong Prime Inheritance in Jordan Systems”. Algebras, Groups and Geom. 1 (1984) 217-234.

[16] K. McCrimmon, “Jordan Centroids”. Comm. Algebra 27 (2) (1999), 933-954.

[17] K. McCrimmon, E. Zelmanov, “The Structure of Strongly Prime Quadratic Jordan Algebras”. Adv. Math. 69 (2) (1988), 133-222.

[18] Yu. A. Medvedev, “An Analogue of Andrunakievich’s Lemma for Jordan Algebras”. Siberian Math. J. 28 (1987) 928-936.

[19] F. Montaner, “Local PI Theory of Jordan Systems II” (to appear).

[20] N. S. Nam, K. McCrimmon, “Minimal Ideals in Quadratic Jordan Algebras”. Proc. Amer. Math. Soc. 88 (4) (1983), 579-583.

[21] V. G. Skosyrskii, “Radicals in Jordan algebras”. Siberian Math. J. 29 (1988) 283-293.

[22] E. I. Zelmanov, “Prime Jordan Triple Systems”. Siberian Math. J. 24 (4) (1983), 509-520.

Keywords: Jordan system, heart, simplicity