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. We also provide examples which show that the results obtained are optimal.

2000 Math. Subj. Class.: 16D30, 16N60, 16W10

0. Introduction and Preliminaries

Some of the most important examples of Jordan systems are those coming from associative systems by symmetrization (cf. [1, 2, 13]). Consequently, there is a long tradition of mathematical work aimed at linking Jordan notions with their corresponding associative ones.

In [14], 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 [3]. 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.

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 [14], and, indeed, the hardest part of the work consists of giving examples that show that those results cannot be improved.

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 pairs and triple systems. This is used in the last section to show that the results of the first section are optimal and do not have pair or triple system analogues.

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

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, the symmetrization R^{(+)} of R is a Jordan algebra, pair, or triple system, respectively, over the same Φ-module R, with products given by U_{x}y = xyx and x^{2} = xx, Q_{x}y = xyx, or P_{x}y = xyx, respectively. We write {x, y, z} for xyz + zyx in the three cases. The reader is referred to [1, 2, 10, 13] 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 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^{(+)} (cf. [13, 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^{+} [10, 1.13, 1.14]

Similarly, one can consider associative versions of the above functors, which are clearly compatible with symmetrizations, Jordan cubes and associative powers.

0.4 Given an associative pair R = (R^{+}, R^{−}) and a ∈ R^{−σ}, the local algebra of R at a is the quotient R_{a}^{σ} = R^{σ(a)}/ Ker a, where R^{σ(a)} is the a-homotope of R, and Ker_{R} a = Ker a = {x ∈ R^{σ} | axa = 0} (cf. [2, 0.4]). Local algebras of an associative algebra or triple system R are simply those of V (R).

The above notions are compatible with the functors V ( ) and T ( ) as well as with symmetrizations (cf. [2, 0.5]).

0.5 Given an associative or Jordan algebra, pair, or triple system R, the heart Heart R of R is the intersection of all nonzero ideals of R.

1. Jordan cubes and associative powers of algebras

Our first result could be obtained from [14, Lemma 3]. However, giving a direct proof is easier 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: Let ≡ denote the congruence modulo I. For any a, x, y ∈ R,

(i) axa = U_{a}x ≡ 0; (ii) aaxy = a{a, x, y} − ayxa ≡ 0, by (i); (iii) xaay = {x, a, a}y − aaxy ≡ 0, by (ii); (iv) xyaa = x{y, a, a} − xaay ≡ 0, by (iii). Thus every associative product of at least 4 factors in which there is a repetition belongs to I. Linearizing, associative products x_{1} ⋯ x_{r} of elements in R are alternating modulo I if r ≥ 4. Thus,

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.

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, hence 2I ≠ 0, and 2I = I, by simplicity.

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 and since R^{5} is a nonzero ideal of R, we get I = R^{5} by simplicity.

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 [12, The First Herstein Construction; 4, 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 (0.4)). Hence, our assertion follows from [7, 3.3].

1.4 Corollary (cf. [3, 4.6 (ii); 4, 4.5]). 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 [4, 4.5] and (1.2).

2. Growing Hearts in Associative Systems

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) and a left primitive algebra acting densely on V on the left.

(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 denumerable direct sum of subspaces V_{i}, all of finite dimension. For any i ∈ N, V = A_{i} ⊕ B_{i}, where

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

It is very easy to see that

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}.

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}.

For any i ∈ N, let f_{i} ∈ End(V) be the projection onto A_{i} relative to the decomposition V = A_{i} ⊕ 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}.

2.2 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 algebras or triple systems. For any 0 ≠ x ∈ R, we have xLx ≠ 0 by [2, 1.7]. 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 inherits primeness from R_{1} by Andrunakievich’s Lemma [3, 4.5]. The pair case follows from the above and [3, 0.3].

2.3 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 and R̂ be its usual unitization.

Let ψ be the diagonal embedding of R into End(R̂^{N}) via the left regular representation, and let τ_{i} be the canonical injection of R̂ into V := R̂^{N}.

Now, S = ψ(R) is a subalgebra of End(V) isomorphic to R, and S∩End(V)_{0} = 0: if 0 ≠ x ∈ R, then, for any i ∈ N, τ_{i}(x) = ψ(x)(τ_{i}(1)) ∈ Im ψ(x), which proves that 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. [5, 0.9(i)]), which is simple and left primitive by (2.1)(i).

\underline{Case 2}: Let R be an associative triple system and R_{1} be an associative algebra such that R is a subsystem of the underlying triple system of R_{1} (cf. [1, 1.13]). Build R̃_{1} out of R_{1} as in Case 1, and let S_{1} denote the subalgebra of R̃_{1} isomorphic to R_{1}. Let S be the homomorphic image of R into S_{1}, so that R̃ := S ⊕ Heart(R̃_{1}) is a subsystem of the underlying triple system of R̃_{1}.

The associative triple system R̃_{1} is prime by [3, 0.3], and R̃ is prime by (2.2). Also, Heart(R̃_{1}) is a simple triple system by [3, 3.1], hence Heart(R̃_{1}) = Heart(R̃)

by [5, 0.9(i)].

Now, Heart(R̃_{1}) = Heart(R̃) is a left primitive triple system by [8, 0.1] and [2, 1.8]. Finally, R̃ is a left primitive triple system by [2, 1.10].

\underline{Case 3}: If R = (R^{+}, R^{−}) is an associative pair, there exists an associative algebra R_{1} such that R is a subpair of V (R_{1}) [11, 2.3], and we can proceed as in Case 2.

2.4 Theorem. Let R be a finite dimensional associative system (algebra, triple system, or pair) over a field Φ. There exists an associative system R̃ and an associative algebra R̃_{1} satisfying (2.3) and (taking R̃_{1} = R̃ in the algebra case) there exist {f_{i} | i ∈ N} ⊆ Heart(R̃_{1}), such that:

(i) 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,

(ii) for any h ∈ Heart(R̃_{1}), there exists i ∈ N such that h = hf_{i} = f_{i}h = f_{i}hf_{i}, (iii) f_{i}s = sf_{i} for any s ∈ S (for any s ∈ S^{σ}, σ = ± in the pair case), i ∈ N.

Proof: Let R be an algebra and let ψ, V, and τ_{i} be as in (2.3). We notice that for any i ∈ N the subspace τ_{i}(R̂) of V is finite dimensional.

For any i ∈ N we take

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

so that

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

is a left ideal of End(V), which is simple and left primitive, acting densely on V on the left by (2.1)(iii). In particular, S ∩ I ⊆ S ∩ End(V)_{0} = 0, as in (2.3).

For any i ∈ N, we also consider the projection f_{i} of V onto A_{i}, so that f_{i} ∈ I. 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} ∘ L_{x})(v_{j})),

which is (τ_{j} ∘ L_{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} ∘ L_{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) (2.1)(ii-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. [5, 0.9(i)]).

Now, (i) and (ii) are just (2.1)(iii)(a) and (b), while (iii) has been already proved. When R is a triple system or a pair, Cases 2 and 3 of (2.3) apply here verbatim.

3. Examples

3.1 Given a countable set of generators {e_{i} | i ∈ A} for some A ⊆ N = {1, 2, …}, let G_{A} denote the algebra generated over 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, i.e., the nonunital Grassmann or exterior algebra on the free module with basis {e_{i} | i ∈ A} (cf. [9, I.7.2, II p. 141]). Thus 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}, whose elements will be called 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 follows that U_{u}v = uvu = 0 whenever u is a monomial, hence the Jordan cube of G_{A} is spanned over Φ by the set {{u, v, w} | u, v, w are monomials}. 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,

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]} = I^{[4]} = 0 since the elements {e_{i}, e_{j}, e_{k}}, {e_{i}e_{j}, e_{k}, e_{l}}, and {e_{i}, e_{j}e_{k}, e_{l}} can be checked to be zero.

Finally, either wvu = uvw or wvu = −uvw for monomials u, v, w, hence

I ⊆ 2G_{A}.  (2)

3.2 Remark: (i) In (1.1) the power 5 is the minimum one can get in general: Take 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 2Φ ≠ 0.

(ii) The factor 2 cannot be removed in (1.1): 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 by (3.1)(2). In particular, when A = N, R satisfies R^{n} ⊈ I for any n ∈ N.

(iii) There is no triple system or pair analogue of (1.1): For any A ⊆ N, let T_{A} = ⨁_{odd n} G_{A}^{[n]}, which is an associative triple system, subsystem of the underlying triple system of 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}. But {u, v, w} = 0 since odd length monomials anticommute, 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, R = T_{A} satisfies 0 ≠ 2R^{n} ⊈ I. Moreover, if A = N, 2R^{n} ⊈ I for any odd n. Taking R = V (T_{A}) provides a pair example.

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 (2.3). The algebra R̃ so obtained is prime with simple heart, and the ideal Ĩ of R̃ generated by (R̃^{(+)})^{3} coincides with Heart(R̃), hence 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: 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 (2.3) 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 or pair 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 (2.3), 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}. If Φ has characteristic not 2, then 2R̃ ≠ 0. Proceeding as above with R = V (T_{A}) provides a pair example.

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 (2.4) are simple.

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

aR̃a = aIa.  (1)

For any b ∈ R̃, we will show that aba ∈ aIa. Indeed, R̃ = S ⊕ I implies a = s + y and b = t + z, for some s, t ∈ S, y, z ∈ I. By (2.4)(i–iii) there is an idempotent e ∈ I of R̃_{1} such that y = ey = ye, et = te, and es = se. Thus 0 = sts = eests = setes and

aba = a(t + z)a = ata + aza = (s + y)t(s + y) + aza = sts + sty + yts + yty + aza =

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

since ete, z ∈ I.

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 [2, 1.7(ii)] since R̃ is prime.

Finally, 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 [3, 3.1] since I is a simple algebra (2.4). Now, I_{a} is simple by [6, 1.5].

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

3.5 Remark: Arguing as in (3.3), for a finite set A, replacing (2.3) by (2.4), and using (3.4), one can obtain examples showing that:

(i) The minimum power n such that Heart(R) = R^{n} in (1.4) is 5.

(ii) If 2R = 0 in (1.4), the heart of R cannot be written in terms of associative powers.

(iii) When dealing with pairs or triple systems, there is no way to describe the heart in terms of associative powers as in (1.4).

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Keywords: Associative system, Jordan system, local algebras, heart, simplicity.