Jordan Cubes and Associative Powers ^{1}

José A. Anquela, Teresa Cortés

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

anque@pinon.ccu.uniovi.es, cortes@pinon.ccu.uniovi.es

Abstract: We study the relations between the powers of an associative algebra and the ideal generated by its Jordan cube. As a consequence, we describe the heart of an associative algebra for which every local algebra is simple in terms of associative powers, answering a question posed in [J. Algebra 240 (2001), 680-704]. We also provide examples which show that the results obtained are optimal.

Introduction

Some of the most important examples of Jordan systems are those coming from associative systems by symmetrization. This shows up in the classifications of Jordan systems under regularity conditions, like strong primeness, primitivity, or simplicity (cf. [1, 3, 4, 10, 18, 21]). Consequently, there is a long tradition of mathematical work aimed at linking Jordan notions with their corresponding associative ones.

In [20], Skosyrskii considers the connections between various nilpotency conditions on right alternative algebras and their corresponding Jordan algebras obtained by symmetrization. He is obviously led to study the relations between the powers of a right alternative algebra and the powers of its symmetrization. The problem appears again in a natural way when studying local-to-global inheritance of simplicity in associative systems [5]. Indeed, it is proved that an associative system for which every local algebra is simple may not be simple but it has a “big” simple heart (cf. [5, 4.1, 4.6 (ii), 4.8 (ii); 6, 4.5, 4.7]), where “big” means equal to the Jordan cube of the system. Here, Jordan products appear due to the “Jordan nature” of the notion of local algebra, in spite of the fact that the setting is purely associative. Thus, it is natural to wonder whether Jordan cubes in the description of the heart can be replaced by associative powers.

This paper is devoted to answering the above question. We will study all possible relations between the ideal I of an associative system R generated by its Jordan cube and the associative powers R^{n}. Under certain conditions, this will also give a description of the heart of R when all of its local algebras are simple. The positive results are based on elementary combinatorial techniques, already appearing in [20], and, indeed, the hardest part of the work consists of giving examples that show that those results cannot be improved.

1 Partially supported by the Ministerio de Ciencia y Tecnología BFM2001-1938-C02-02 and the Universidad de Oviedo AYP-01-06

Fax number: ++34 985 102 886

After a preliminary section, we begin in Section 1 by showing that the ideal I mentioned above always contains 2R^{5} when R is an algebra. As a consequence, an associative algebra R such that every one of its local algebras is simple has Heart(R) = R^{5} or 2R = 0. In the second section, we show how to grow nonzero hearts inside every associative algebra. The constructions, based on the regular representation, can be extended to the cases of associative pairs and triple systems. This is used in the last section of the paper to build examples which show that the results for algebras of the first section are the best possible ones and do not have pair or triple system analogues.

0. Preliminaries

0.1 We will deal with associative algebras, pairs, and triple systems over an arbitrary ring of scalars Φ. Given an associative algebra, pair, or triple system R, we denote by R^{(+)} its symmetrization: over the Φ-module R we consider the Jordan products U_{x}y, Q_{x}y or P_{x}y equal to the associative products xyx, whose linearization xyz + zyx will be denoted by {x, y, z}, and the Jordan squares x^{2} (in the algebra case), which are just the associative squares xx. The reader is referred to [1, 4, 12, 15, 18] for basic facts and notation regarding Jordan and associative systems.

0.2 Given an associative algebra or triple system R, its Jordan cube (R^{(+)})^{3} is just the linear span of {xyx | x, y ∈ R}, while the associative power R^{n} for a natural number n (odd in the triple system case) is the linear span of {x_{1} ⋯ x_{n} | x_{1}, …, x_{n} ∈ R}.

Similarly, if R = (R^{+}, R^{−}) is an associative pair, its Jordan cube (R^{(+)})^{3} is the pair (Q_{R^{+}}R^{−}, Q_{R^{−}}R^{+}), where Q_{R^{σ}}R^{−σ} is the linear span of {xyx | x ∈ R^{σ}, y ∈ R^{−σ}}, σ = ±. The associative power R^{n} for an odd natural number n is the pair (S^{+}, S^{−}), where S^{σ} is the linear span of {x_{1} ⋯ x_{n} | x_{1}, x_{3}, …, x_{n} ∈ R^{σ}, x_{2}, x_{4}, …, x_{n−1} ∈ R^{−σ}}, σ = ±.

Clearly, in all cases R^{n} is an associative ideal of R. The Jordan cube (R^{(+)})^{3} is an ideal of R^{(+)} (the assertion is obvious in the pair and triple system cases, whereas for algebras it follows from [18, 0.14]).

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^{+} [15, 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.

The above functors are clearly compatible with symmetrizations, Jordan cubes and associative powers.

0.4 Regularity conditions through the functors V and T [5, 0.3, 3.1, 3.4; 1, p. 230; 9, 1.12]:

(i) Let T be a Jordan (resp. associative) triple system.

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

(b) If V (T) is strongly prime (resp. prime), then T is strongly prime (resp. prime),

(c) If V (T) is simple, then T is simple.

(ii) Let V be a Jordan (resp. associative) pair.

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

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

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

(iii) Let J be a Jordan (resp. associative) algebra.

(a) J is nondegenerate (resp. semiprime) if and only if V (J) is nondegenerate (resp. semiprime),

(b) J is strongly prime (resp. prime) if and only if V (J) is strongly prime (resp. prime),

(c) J is simple if and only if V (J) is simple.

0.5 Local algebras, introduced in [19], are one of the most useful tools to study primitivity of Jordan pairs and triple systems [2, 4]. They are the way to connect the categories of algebras and pairs and triple systems:

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

x ·_{a} y = xay,

for any x, y ∈ R^{σ}. The set

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

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

R_{a}^{σ} = R^{σ(a)}/ Ker a

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

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

The above notions are compatible with the functors V ( ) and T ( ) as well as with symmetrizations (cf. [4, 0.5]).

1. Jordan cubes and associative powers of algebras

Our first result could be obtained from [20, Lemma 3]. However, due to the more general setting considered in [20], applying [20, Lemma 3] would be more cumbersome than giving a direct proof, very simple in our setting.

1.1 Proposition. For any associative algebra R, the ideal I of R generated by the Jordan cube (R^{(+)})^{3} contains 2R^{5}.

Proof: Denote by ≡ the congruence modulo I. We claim that every associative product of at least 4 factors in which there is a repetition belong to I. Let us consider such a product of elements a, x, y, … ∈ R, where a appears twice.

(i) If the repeated factors are not consecutive, then

⋯ ax ⋯ a ⋯ = ⋯ U_{a}(x ⋯) ⋯ ∈ I.

(ii) If the repeated factors are consecutive, they can appear

- at the beginning of the product, and then

aaxy ⋯ = a{a, x, y} ⋯ − ayxa ⋯ ≡ 0,

by (i); - in the middle of the product, and then

⋯ xaay ⋯ = ⋯ {x, a, a}y ⋯ − ⋯ aaxy ⋯ ≡ 0,

by the previous case;

- at the end of the product, and then

⋯ xyaa = ⋯ x{y, a, a} − ⋯ xaay ≡ 0,

by the previous case.

Linearizing, we obtain that every associative product x_{1} ⋯ x_{r} of elements in R is alternating modulo I if r ≥ 4. Thus, for a product of exactly 5 elements of R, we have:

x_{1}x_{2}x_{3}x_{4}x_{5} = x_{1}x_{2}(x_{3}x_{4})x_{5} ≡ −x_{1}x_{2}x_{5}(x_{3}x_{4}) =
−x_{1}x_{2}x_{5}x_{3}x_{4} ≡ x_{1}x_{2}x_{3}x_{5}x_{4} ≡ −x_{1}x_{2}x_{3}x_{4}x_{5},

hence 2x_{1}x_{2}x_{3}x_{4}x_{5} ≡ 0, and therefore 2R^{5} ⊆ I.

When the ideal I of R as in (1.1) is simple and R is prime, we can get more than a containment.

1.2 Theorem. Let R be a prime associative algebra such that the ideal I of R generated by the Jordan cube (R^{(+)})^{3} is simple. Then, either I = R^{5} or 2R = 0.

Proof: In general, the kernel Ker(2Id_{R}) of 2Id_{R} and 2R are orthogonal ideals of R. If 2R ≠ 0, the primeness of R implies Ker(2Id_{R}) = 0.

Thus 2I is a nonzero ideal of I, hence 2I = I, by simplicity of I.

For any x ∈ R^{5}, 2x ∈ I by (1.1). Hence, I = 2I implies that there exists y ∈ I such that 2x = 2y. Thus, x − y ∈ Ker(2Id_{R}) = 0, and x = y. This shows R^{5} ⊆ I. Moreover, since R^{5} is a nonzero ideal of R, it is a nonzero ideal of I, and I = R^{5} by simplicity of I.

As a consequence of the previous result, we will get an improved version of [5, 4.6 (ii); 6, 4.5] in the case of algebras. Before that, we need a local characterization of primeness of associative algebras, i. e., an associative version of [9, 3.3]. The proof is based on the Jordan result [9, 3.3] and the transfer of primeness through symmetrizations. An independent associative proof would just consist of a tedious repetition of Jordan arguments, unavoidable due to the Jordan nature of local algebras.

1.3 Lemma. Let R be a nonzero associative algebra. Then R is prime if and only if R_{a} is prime and nonzero, for any 0 ≠ a ∈ R.

Proof: By [17, The First Herstein Construction; 6, 1.2 (ii)], R is prime if and only if R^{(+)} is strongly prime. Also, R_{a} is prime if and only if (R^{(+)})_{a} is strongly prime, since (R^{(+)})_{a} = R_{a}^{(+)} (see (0.5)). Hence, our assertion follows from [9, 3.3].

1.4 Corollary. Let R be an associative algebra such that the local algebra R_{a} is simple for any 0 ≠ a ∈ R. Then, the heart Heart R = (R^{(+)})^{3} is simple. Moreover, if 2R ≠ 0, then Heart R = R^{5}.

Proof: Any simple algebra is prime and nonzero, so that R is prime using (1.3). Thus, the result follows from [6, 4.5] and (1.2).

2. Growing Hearts in Associative Systems

The aim of this section is pasting a simple heart to any associative system. Thus we begin by recalling some examples of simple systems of endomorphisms.

2.1 (i) Let Φ be a field and let V be a vector space over Φ. It is wellknown that the set End(V)_{0} of all endomorphisms of V of finite rank is a simple ideal of End(V). Moreover, it is straightforward that End(V)_{0} acts densely on V on the left, hence it is also a left primitive associative algebra.

(ii) Let Φ be a field, V be a vector space over Φ, and B = {v_{i} | i ∈ I} be a basis of V over Φ. For any i, j ∈ I, let e_{ij} ∈ End(V) be given by e_{ij}(v_{k}) = δ_{jk}v_{i}. The linear span End(V)_{B} of {e_{ij} | i, j ∈ I} is contained in End(V)_{0}. It is readily seen that the endomorphisms e_{ij} behave like usual matrix units:

e_{ij}e_{kl} = δ_{jk}e_{il}, for any i, j, k, l ∈ I.

Consequently, End(V)_{B} is a simple subalgebra of End(V). Moreover, for any f ∈ End(V), if f(v_{i}) = ∑_{k∈I} α_{k}v_{k}, then

fe_{ij} = ∑_{k∈I} α_{k}e_{kj},

which implies that End(V)_{B} is a left ideal of End(V). Also End(V)_{B} is left primitive, since it acts densely on V on the left.

(iii) Let Φ be a field and let V = ⨁_{i∈N} V_{i} a vector space which is a direct sum of a denumerable amount of subspaces V_{i}, all of which have finite dimension. For any i ∈ N, define:

A_{i} = ⨁_{k≤i} V_{k}, B_{i} = ⨁_{k>i} V_{k}.

Clearly, V = A_{i} ⊕ B_{i} for any i ∈ N. Let

I = {f ∈ End(V) | B_{i} ⊆ Ker f, for some i ∈ N}.

It is very easy to see that

I = {f ∈ End(V) | B_{i} ⊆ Ker f and A_{i} ⊇ Im f, for some i ∈ N}.

Choosing a basis G_{i} of V_{i} for each i ∈ N and taking the basis B = ∪_{i∈N}G_{i} of V, it is straightforward to check that

I = End(V)_{B}.

Thus, I is a left ideal of End(V), it is simple and left primitive, acting densely on V on the left.

For any i ∈ N, let f_{i} ∈ End(V) be given by f_{i}(v) = v for any v ∈ A_{i} and f_{i}(v) = 0 for any v ∈ B_{i}. It is obvious that f_{i} ∈ I and:

(a) f_{i}f_{j} = f_{j}f_{i} = f_{min{i,j}} for any i, j ∈ N and, in particular, f_{i} is an idempotent for any i ∈ N.

(b) For any h ∈ I, there exists i ∈ N such that h = hf_{i} = f_{i}h = f_{i}hf_{i}.

Our next goal is growing a simple primitive heart of the form End(V)_{0} inside an arbitrary associative system. To do that, we need a lemma to transfer primeness to sufficiently big subsystems.

2.2 Recall that a nonzero ideal of a prime associative system is prime. When dealing with algebras this is immediately obtained by using Andrunakievich’s Lemma. If we consider an ideal L of a prime triple system R, we can use an Andrunakievichlike result too: For any nonzero ideal K of L, the ideal of R generated by KKKKK is contained in K (cf. [5, 4.5]) and it is nonzero by semiprimeness of L, which is obtained from semiprimeness of R using [4, 1.7(i)]. Finally, the pair case follows from the triple system case through the functor T ( ) by (0.4)(ii)(b).

2.3 Lemma. Let R_{1} be a prime associative system (algebra, triple system, or pair), and R be a subsystem of R_{1} containing a nonzero ideal L of R_{1}. Then R is prime.

Proof: Let us assume that we deal with triple systems. For any 0 ≠ x ∈ R, we have xLx ≠ 0 by [4, 1.7(ii)]. This readily implies that any nonzero ideal of R hits L. Thus, nonzero orthogonal ideals of R would give rise to nonzero orthogonal ideals of L, which is impossible since L is prime by (2.2), showing that R is prime.

The cases of algebras and pairs follow from the triple system case by using (0.4). 2.4 Theorem. Let R be an associative system (algebra, triple system, or pair)

over a field Φ. There exists an associative system R̃ such that:

(i) R is isomorphic to a subsystem S of R̃, (ii) R̃ is a left primitive system, hence it is prime, (iii) Heart(R̃) is simple and left primitive, (iv) R̃ = S ⊕ Heart(R̃), hence R̃/ Heart(R̃) ≅ R. Moreover,

(a) if R is an associative triple system, then Heart(R̃) = Heart(R̃_{1}), for a left primitive associative algebra R̃_{1} such that R̃ is a subsystem of the underlying triple system of R̃_{1}, and Heart(R̃_{1}) is simple and left primitive as an algebra;

(b) if R is an associative pair, then Heart(R̃) = V (Heart(R̃_{1})), for a left primitive associative algebra R̃_{1} such that R̃ is a subpair of V (R̃_{1}), and Heart(R̃_{1}) is simple and left primitive as an algebra. Proof: \underline{Case 1}: Let R be an associative algebra. Consider the regular left

representation of R in End(R̂), where R̂ is the usual unitization of R:

φ: R ⟶ End(R̂)

x ↦ φ(x) = L_{x},

where L_{x}(y) = xy for any x ∈ R, y ∈ R̂. It is obvious that φ is a monomorphism of algebras.

Take a external direct sum of a denumerable amount of copies of R̂,

V = ∑_{i∈N} V_{i}, V_{i} = R̂, for any i ∈ N.

We denote by τ_{i} the canonical injection from V_{i} into V, so that V is the internal direct sum V = ⨁_{i∈N} τ_{i}(V_{i}).

We define

ψ: R ⟶ End(V)
x ↦ ψ(x) = ⊕_{i∈N}(τ_{i} ∘ φ(x)),

i. e., ψ(x) is the unique element in End(V) such that ψ(x) ∘ τ_{i} = τ_{i} ∘ φ(x) for any i ∈ N. It is easy to prove that ψ is a monomorphism of algebras.

Now, S = ψ(R) is a subalgebra of End(V) isomorphic to R.

We claim that S ∩ End(V)_{0} = 0. Indeed, if 0 ≠ x ∈ R, then, for any i ∈ N, τ_{i}(x) = τ_{i}(φ(x)(1)) = ψ(x)(τ_{i}(1)) ∈ Im ψ(x), which proves that ψ(x) does not have finite rank, that is to say, ψ(x) ∉ End(V)_{0}.

Take R̃ = S + End(V)_{0}, which is a direct sum of a subalgebra and an ideal of End(V) (see (2.1)(i)), hence it is a subalgebra of End(V). Moreover, End(V)_{0} is an ideal of R̃. Since End(V)_{0} acts densely on the left on V, so does R̃, hence R̃ is left primitive and, in particular, it is a prime algebra. Now End(V)_{0} is a simple ideal of a prime algebra, so that it is its heart, Heart(R̃) = End(V)_{0} (cf. [7, 0.9(i)]), which is simple and left primitive by (2.1)(i).

\underline{Case 2}: Let R be an associative triple system. By [1, 1.13], there exists an associative algebra R_{1} over Φ such that R is a subsystem of the underlying triple system of R_{1}. By the previous case, there exists a left primitive associative algebra R̃_{1} such that R_{1} is isomorphic to a subalgebra S_{1} of R̃_{1}, R̃_{1} = S_{1} ⊕ Heart(R̃_{1}), and Heart(R̃_{1}) is a simple left primitive algebra. Let S be the homomorphic image of R into S_{1}, so that S is a subsystem of the underlying triple system of R̃_{1}. Since Heart(R̃_{1}) is an algebra ideal of R̃_{1}, it is also an ideal of the underlying triple system of R̃_{1}, so that

R̃ = S ⊕ Heart(R̃_{1})

is a subsystem of the underlying triple system of R̃_{1}. Clearly S is a subsystem of R̃ isomorphic to R.

The associative system R̃_{1} is prime as an algebra since it is left primitive, hence it is prime as a triple system by (0.4). Thus, R̃ is prime by (2.3).

Moreover, Heart(R̃_{1}) is a simple algebra, hence it is simple as a triple system by (0.4). Since it is an ideal of R̃, which is prime, we obtain

Heart(R̃_{1}) = Heart(R̃)

by [7, 0.9(i)].

Now, Heart(R̃_{1}) = Heart(R̃) is a left primitive algebra and then it is a left primitive triple system by [11, 0.1] and [4, 1.8(ii)(a)].

Finally, R̃ is a left primitive triple system by [4, 1.10(ii)(b)] since it is prime and has a left primitive nonzero ideal.

\underline{Case 3}: Let R = (R^{+}, R^{−}) be an associative pair. By [16, 2.3], there exists an associative algebra R_{1} over Φ such that R is a subpair of V (R_{1}). By the first case, there exists a left primitive associative algebra R̃_{1} such that R_{1} is isomorphic to a subalgebra S_{1} of R̃_{1}, R̃_{1} = S_{1} ⊕ Heart(R̃_{1}), and Heart(R̃_{1}) is a simple left primitive algebra. Let S be the homomorphic image of R into V (S_{1}), so that S is a subpair of V (R̃_{1}). Since Heart(R̃_{1}) is an algebra ideal of R̃_{1}, V (Heart(R̃_{1})) is also an ideal of V (R̃_{1}), so that

R̃ = S ⊕ V (Heart(R̃_{1}))

is a subpair of V (R̃_{1}). Clearly S is a subpair of R̃ isomorphic to R.

The associative system R̃_{1} is prime as an algebra since it is left primitive, hence V (R̃_{1}) is prime by (0.4). Thus, R̃ is prime by (2.3).

Moreover, Heart(R̃_{1}) is a simple algebra, hence V (Heart(R̃_{1})) is simple by (0.4). Since it is an ideal of R̃, which is prime, we have

V (Heart(R̃_{1})) = Heart(R̃)

by [7, 0.9(i)].

Now, Heart(R̃_{1}) is a left primitive algebra, hence V (Heart(R̃_{1})) = Heart(R̃) is a left primitive pair by [11, 0.1] and [4, 1.8(i)].

Finally, R̃ is a left primitive pair by [4, 1.10(i)(b)] since it is prime and has a left primitive nonzero ideal.

When the associative system is finite dimensional over the field Φ we can grow a heart as above with the extra advantage of having nice behaved idempotents.

2.5 Theorem. Let R be a finite dimensional associative system (algebra, triple system, or pair) over a field Φ. There exists an associative system R̃ such that:

(i) R is isomorphic to a subsystem S of R̃, (ii) R̃ is a left primitive system, hence it is prime, (iii) Heart(R̃) is simple and left primitive, (iv) R̃ = S ⊕ Heart(R̃), hence R̃/ Heart(R̃) ≅ R. Moreover,

(a) if R is an associative triple system, then Heart(R̃) = Heart(R̃_{1}), for a left primitive associative algebra R̃_{1} such that R̃ is a subsystem of the underlying triple system of R̃_{1}, and Heart(R̃_{1}) is simple and left primitive as an algebra;

(b) if R is an associative pair, then Heart(R̃) = V (Heart(R̃_{1})), for a left primitive associative algebra R̃_{1} such that R̃ is a subpair of V (R̃_{1}), and Heart(R̃_{1}) is simple and left primitive as an algebra. In all cases (taking R̃_{1} = R̃ in the algebra case) there exist {f_{i} | i ∈ N} ⊆

Heart(R̃_{1}), such that:

(v) f_{i}f_{j} = f_{j}f_{i} = f_{min{i,j}} for any i, j ∈ N and, in particular, f_{i} is an algebra idempotent for any i ∈ N, (vi) for any h ∈ Heart(R̃_{1}), there exists i ∈ N such that h = hf_{i} = f_{i}h = f_{i}hf_{i},

(vii) f_{i}s = sf_{i} for any s ∈ S (for any s ∈ S^{σ}, σ = ± in the pair case), i ∈ N.

Proof: Assume first that R is an algebra. As in the first case of the proof of (2.4), we take a external direct sum of a denumerable amount of copies of R̂,

V = ∑_{i∈N} V_{i}, V_{i} = R̂, for any i ∈ N.

We denote by τ_{i} the canonical injection from V_{i} into V, so that V is the internal direct sum V = ⨁_{i∈N} τ_{i}(V_{i}). Notice that in this case the subspaces τ(V_{i}) of V are finite dimensional.

We also have

ψ: R ⟶ End(V)
x ↦ ψ(x) = ⊕_{i∈N}(τ_{i} ∘ φ(x)),

i. e., ψ(x) is the unique element in End(V) such that ψ(x) ∘ τ_{i} = τ_{i} ∘ φ(x) for any i ∈ N, where φ is the regular representation of R in End(R̂). We remark that ψ is a monomorphism of algebras.

Now, S = ψ(R) is a subalgebra of End(V) isomorphic to R.

As in (2.1)(iii), for any i ∈ N, define:

A_{i} = ⨁_{k≤i} τ_{k}(V_{k}), B_{i} = ⨁_{k>i} τ_{k}(V_{k}).

Then

I := {f ∈ End(V) | B_{i} ⊆ Ker f, for some i ∈ N} =
{f ∈ End(V) | B_{i} ⊆ Ker f and A_{i} ⊇ Im f, for some i ∈ N} ⊆ End(V)_{0},

and it is a left ideal of End(V), which is simple and left primitive, acting densely on V on the left.

In particular, S ∩ I ⊆ S ∩ End(V)_{0} = 0, as in (2.4).

For any i ∈ N, we also consider f_{i} ∈ I given by f(v) = v for any v ∈ A_{i} and f(v) = 0 for any v ∈ B_{i}, as in (2.1)(iii).

We claim that f_{i}s = sf_{i}, for any s ∈ S and any i ∈ N. Indeed, s = ψ(x) for some x ∈ R, hence, for any j ∈ N, v_{j} ∈ V_{j},

f_{i}s(τ_{j}(v_{j})) = (f_{i} ∘ ψ(x))(τ_{j}(v_{j})) = f_{i}((ψ(x) ∘ τ_{j})(v_{j})) = f_{i}((τ_{j} ∘ φ(x))(v_{j})),

which is (τ_{j} ∘ φ(x))(v_{j}) if j ≤ i, 0 otherwise, and

sf_{i}(τ_{j}(v_{j})) = (ψ(x) ∘ f_{i})(τ_{j}(v_{j}))

which is ψ(x)(τ_{j}(v_{j})) = (ψ(x) ∘ τ_{j})(v_{j}) = (τ_{j} ∘ φ(x))(v_{j}) if j ≤ i, 0 otherwise.

Take R̃ = S + I, which is a direct sum of a subalgebra and a left ideal of End(V) (see (2.1)(iii)). Moreover, IS ⊆ I: for any h ∈ I, there exists i ∈ N such that h = hf_{i} by (2.1)(iii)(b), so that for any s ∈ S we have hs = hf_{i}s = hsf_{i} = h(sf_{i}) ∈ II ⊆ I since I is a left ideal of End(V). This proves that R̃ is a subalgebra of End(V) and that I is an ideal of R̃. Since I acts densely on the left on V, so does R̃, hence R̃ is left primitive and, in particular, it is a prime algebra. Now I is a simple ideal of a prime algebra, so that it is its heart, Heart(R̃) = I (cf. [7, 0.9(i)]).

Finally, (v) and (vi) are just (2.1)(iii)(a) and (b), while (vii) has been already proved.

When R is a triple system or a pair, Cases 2 and 3 of the proof of (2.4) apply here verbatim.

3. Examples

We give a list of examples which show that the results of the first section cannot be improved in the case of algebras and have no possible analogue in the cases of pairs and triple systems.

3.1 The Grassmann Algebra: Given a countable set of generators {e_{i} | i ∈ A} for some A ⊆ N = {1, 2, …}, the Grassmann algebra G_{A} is the algebra generated over a ring of scalars Φ by the set {e_{i} | i ∈ A} with the relations e_{i}^{2} = 0 and e_{i}e_{j} = −e_{j}e_{i} for any i, j ∈ A. It can be shown that G_{A} is a free Φ-module with basis {e_{i_{1}} ⋯ e_{i_{r}} | i_{1} < … < i_{r}, i_{j} ∈ A for any j = 1 … r} (cf. [13, 7.2; 14, p. 141]). From now on we will refer to the elements of this basis as monomials. Recall that G_{A} has a natural grading over N, G_{A} = ⨁_{n∈N} G_{A}^{[n]}, where G_{A}^{[n]} is the Φ-module generated by the set {e_{i_{1}} ⋯ e_{i_{n}} | i_{1} < … < i_{n}, i_{j} ∈ A for any j = 1 … n}. The elements of G_{A}^{[n]} will be said of length n.

From e_{i}^{2} = 0 and e_{i}e_{j} = −e_{j}e_{i} for any i, j ∈ A it easily follows that the Jordan product U_{u}v = uvu = 0 whenever u is a monomial, hence the Jordan cube of G_{A} is generated as Φ-module by the Jordan products {{u, v, w} | u, v, w are monomials}. These are homogeneous elements with respect to the above grading, so that the ideal I of G_{A} generated by its Jordan cube is graded, I = ⨁_{n∈N} I^{[n]}, where I^{[n]} = I ∩G_{A}^{[n]} for any n ∈ N. Moreover we claim that

I^{[n]} = 0 for n = 1, 2, 3, 4.  (1)

Indeed, this is true for n = 1, 2 because the generators of the ideal I have at least length 3. On the other hand, I^{[3]} is generated as a Φ-module by the elements {e_{i}, e_{j}, e_{k}} = e_{i}e_{j}e_{k} + e_{k}e_{j}e_{i} and then the equality e_{k}e_{j}e_{i} = −e_{k}e_{i}e_{j} = e_{i}e_{k}e_{j} = −e_{i}e_{j}e_{k} implies I^{[3]} = 0. Now, I^{[4]} is generated as a Φ-module by the elements {e_{i}e_{j}, e_{k}, e_{l}} and {e_{i}, e_{j}e_{k}, e_{l}}. But e_{l}e_{k}e_{i}e_{j} = e_{l}e_{i}e_{j}e_{k} = e_{i}e_{j}e_{l}e_{k} = −e_{i}e_{j}e_{k}e_{l}, which means {e_{i}e_{j}, e_{k}, e_{l}} = 0. Analogously, {e_{i}, e_{j}e_{k}, e_{l}} = 0, hence I^{[4]} = 0.

Finally, we notice that one always has either wvu = uvw or wvu = −uvw for monomials u, v, w, hence it follows that

I ⊆ 2G_{A}.  (2)

3.2 Remark: (i) In (1.1) the power 5 is the minimum one can get in general. This can be seen taking A = {1, 2, 3, 4}, so that R = G_{A} has I = 0 by (3.1)(1) while 2R^{4} contains 2e_{1}e_{2}e_{3}e_{4} ≠ 0 if Φ is, for example, a field of characteristic not 2.

(ii) The factor 2 cannot be removed in (1.1). Indeed, for any A ⊆ N with at least n elements, and any Φ such that \frac{1}{2} ∉ Φ, the algebra R = G_{A} has 0 ≠ e_{i_{1}} ⋯ e_{i_{n}} ∈ R^{n} \ I since I ⊆ 2R by (3.1)(2). In particular, when A = N, R satisfies R^{n} ⊈ I for any n ∈ N.

(iii) There is no triple system analogue of (1.1): Consider, for any A ⊆ N, T_{A} = ⨁_{odd n} G_{A}^{[n]}, which is an associative triple system, subsystem of the underlying triple system of the algebra G_{A}. As in (3.1) the Jordan cube of T_{A} is spanned over Φ by the set {{u, v, w} | u, v, w are monomials of odd length}. The defining relations of G_{A} readily imply that uv = −vu for any pair of monomials u, v of odd length, so that wvu = −vwu = vuw = −uvw imply {u, v, w} = 0 for such monomials, and the ideal I of T_{A} generated by its Jordan cube is zero. Taking A ⊆ N with at least n elements (n odd) and 2Φ ≠ 0, we get an associative triple system R = T_{A} such that 0 ≠ 2R^{n} ⊈ I. Moreover, if A = N, R satisfies 2R^{n} ⊈ I for any odd n.

(iv) There is no pair analogue of (1.1). To illustrate this assertion one just need to take R = V (T_{A}) for T_{A} as in (iii).

The examples in (3.2) satisfy that the ideal generated by the Jordan cube is zero. To evaluate (1.2), we will grow simple hearts by using (2.4).

3.3 Remark: (i) The power 5 is the minimum satisfying the assertion of (1.2): Take R as in (3.2)(i) over a field Φ and apply (2.4). The algebra R̃ so obtained is prime and has 0 ≠ Heart(R̃) simple. Moreover, R̃ = S ⊕ Heart(R̃), with S ≅ R. Thus, the Jordan cube (R̃^{(+)})^{3} is contained in (S^{(+)})^{3} + Heart(R̃) = Heart(R̃) since (S^{(+)})^{3} ≅ (R^{(+)})^{3} = 0. Hence, the ideal Ĩ of R̃ generated by (R̃^{(+)})^{3} is contained in Heart(R̃). On the other hand, Ĩ ≠ 0 since (R̃^{(+)})^{3} ≠ 0 by the semiprimeness of R̃, and, therefore, Ĩ = Heart(R̃), which is simple. However, R̃^{4} ⊇ S^{4} ≅ R^{4} ≠ 0, S^{4} ⊆ S, and S ∩ Ĩ = S ∩ Heart(R̃) = 0 imply R̃^{4} ⊈ Ĩ. If the characteristic of Φ is not 2 we also have 2R̃ ≠ 0.

(ii) Under the conditions of (1.2), no information on the powers of R and the ideal generated by its Jordan cube can be obtained if 2R = 0. Indeed, if we take in (3.2)(ii) A = N and a field Φ of characteristic 2 we have an algebra R such that R^{n} ≠ 0 for all n ∈ N while the ideal I generated by (R^{(+)})^{3} is zero. Using (2.4) and arguing as in (i), we obtain a prime algebra R̃ whose ideal Ĩ generated by its Jordan cube is simple and does not contain any power R̃^{n}.

(iii) There is no triple system analogue of (1.2): The example R = T_{A} of (3.2)(iii) over a field Φ, with A = N, satisfies that the ideal I of R generated by its Jordan cube is zero, while R^{n} ≠ 0 for all odd n. By (2.4), arguing as in (i), we get a prime associative triple system R̃ whose ideal Ĩ generated by its Jordan cube is simple and does not contain any power R̃^{n}. When Φ has characteristic different from 2 we have 2R̃ ≠ 0 too.

(iv) In the pair setting, there is no analogue of (1.2). Here we cannot proceed as in (3.2)(iv) since the functor V ( ) does not preserve primeness in general. However, to find suitable examples we can proceed as in (iii), beginning with R = V (T_{A}).

In the examples above it is hard to tell whether all local algebras are simple. This is the reason we gave a second way to grow hearts (2.5), this time in finite dimensional systems, so that we can obtain counterexamples to all possible generalizations of (1.4).

3.4 Lemma. Let R be a finite dimensional associative system over a field Φ such that its Jordan cube (R^{(+)})^{3} is zero. Then, all local algebras at nonzero elements of the system R̃ found in (2.5) are simple.

Proof: Let R be an associative algebra or triple system and let 0 ≠ a ∈ R̃. If we denote I = Heart(R̃), we claim that

aR̃a = aIa.  (1)

Indeed, since R̃ = S ⊕ I we can write a = s + y, for some s ∈ S, y ∈ I. For any b ∈ R̃ we will show that aba ∈ aIa. Since b = t + z, for some t ∈ S and z ∈ I, aba = ata + aza, where aza ∈ aIa. On the other hand,

ata = (s + y)t(s + y) = sts + sty + yts + yty,  (2)

where sts = 0 since (S^{(+)})^{3} ≅ (R^{(+)})^{3} = 0. By (2.5)(v–vii) there exists an idempotent e ∈ I of the algebra R̃_{1} such that y = ey = ye, et = te, and es = se. Then

— sts = 0 ⇒ sts = estse = setes,

— sty = stey = sety = setey,

— yts = yets = ytes = yetes,

— yty = yetey.

Now, replacing the summands of (2), we obtain

ata = setes + setey + yetes + yetey = (s + y)ete(s + y) = aetea ∈ aIa

since e ∈ I. Notice that even length products make sense in the triple case too since they can be performed in the algebra R̃_{1} (cf. (2.5)(a)).

By (1), R̃ = I + KerR̃ a, hence

R̃_{a} = R̃/ KerR̃ a = (I + KerR̃ a)/ KerR̃ a ≅ I/(I ∩ KerR̃ a) = I/ Ker_{I} a = I_{a}.

We remark that I_{a} is nonzero by [4, 1.7(ii)] since R̃ is prime.

Finally, due to the wellknown non-inheritance of regularity conditions by local algebras of triple systems (cf. [5, 2.6]), we go to a pair setting to obtain the simplicity of I_{a}. For this, we notice that I_{a} is a generalized local algebra of the pair V (I), which is an ideal of V (R̃_{1}) and it is a simple pair by (0.4)(iii)(c) since I is a simple algebra (2.5)(iii). Now, I_{a} is simple by [8, 1.5].

The above proof is also valid in the pair case with obvious changes.

3.5 Remark: (i) The minimum power n such that Heart(R) = R^{n} in (1.4) is 5. To see that, take R as in (3.2)(i) over a field Φ and apply (2.5) to get the algebra R̃. Since R is finite dimensional and has zero Jordan cube (see (3.2)(i)), all local algebras at nonzero elements of R̃ are simple by (3.4). Arguing as in (3.3)(i) we have R̃^{4} ⊈ Heart(R̃), and 2R̃ ≠ 0 if the characteristic of Φ is not 2.

(ii) If 2R = 0 in (1.4), the heart of R cannot be written in terms of associative powers: Take R = G_{A} for A = {1, …, n} over a field Φ of characteristic 2. As in (3.3)(ii), the Jordan cube of R is zero, hence if we apply (2.5) to R, we obtain an algebra R̃ for which all local algebras at nonzero elements are simple by (3.4). Moreover, R̃ = S ⊕ Heart(R̃), where S is isomorphic to R. Hence R^{n} ≠ 0 (see (3.2)(ii)) implies R̃^{n} ⊇ S^{n} ≠ 0, so that R̃^{n} ⊈ Heart(R̃).

(iii) When dealing with triple systems, there is no way to describe the heart in terms of associative powers as in (1.4). Indeed, the associative triple system R = T_{A} of (3.2)(iii) over a field Φ, with A = {1, …, n} for odd n, satisfies that the ideal I of R generated by its Jordan cube is zero, while R^{n} ≠ 0. Applying (2.5) we get by (3.4) an associative triple system R̃ all of whose local algebras at nonzero elements are simple. As in (ii), it is easily seen that R̃^{n} ⊈ Heart(R̃).

(iv) The description of the heart in terms of associative powers of (1.4) cannot be extended to the pair setting. This can be seen by proceeding as in (iii) above, taking the associative pair R = V (T_{A}).

REFERENCES

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

[2] A. D’Amour, K. McCrimmon, “The Local Algebras of Jordan Systems”. J. Algebra 177 (1995), 199-239.

[3] A. D’Amour, K. McCrimmon, “The Structure of Quadratic Jordan Systems of Clifford Type” (to appear).

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

[5] J. A. Anquela, T. Cortés, “Local and Subquotient Inheritance of Simplicity in Jordan Systems”. J. Algebra 240 (2001), 680-704.

[6] J. A. Anquela, T. Cortés, E. García, “Herstein’s Theorems and Simplicity of Hermitian Jordan Systems”. J. Algebra 246 (2001), 193-214.

[7] J. A. Anquela, T. Cortés, E. García, “Simplicity of the Heart of a Nondegenerate Jordan System”. J. Pure Appl. Algebra (to appear).

[8] J. A. Anquela, T. Cortés, E. García, “Local and Subquotient Inheritance of the Heart of a Jordan System”. Manuscripta Math. (to appear).

[9] 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.

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

[11] J. A. Anquela, T. Cortés, F. Montaner, “Local Inheritance in Jordan Algebras”. Arch. Math. 64 (1995), 393-401.

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

[13] N. Jacobson, Basic Algebra I, W. H. Freeman and Company, New York, 1985.

[14] N. Jacobson, Basic Algebra II, W. H. Freeman and Company, New York, 1989.

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

[16] O. Loos, “Elementary Groups and Stability for Jordan Pairs”. K-Theory 9 (1995), 77-116.

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

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

[19] K. Meyberg. Lectures on Algebras and Triple Systems. Lecture Notes, University of Virginia, Charlottesville, 1972.

[20] V. G. Skosyrskii, “Nilpotency in Jordan and Right Alternative Algebras”. Algebra and Logic 18 (1) (1979) 49-56.

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