Minimal Ideals of Jordan Systems ^{1}

José A. Anquela, Teresa Cortés

anque@orion.ciencias.uniovi.es, cortes@orion.ciencias.uniovi.es Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain

Abstract: We show that minimal ideals of Jordan systems (algebras, triple systems, and pairs) are either simple or trivial, so answering the question posed by Nam and McCrimmon in 1983.

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

0. Introduction and Preliminaries

Minimal ideals, in particular, the heart, appear naturally when studying Jordan systems [10, III.6]. Indeed, having a nonzero heart can be considered as a strong version of primeness for semiprime or nondegenerate systems.

Using Andrunakievich’s Lemma, it is very easy to prove that a minimal ideal of an associative algebra is either simple or trivial. However, a Jordan version of Andrunakievich’s Lemma is false even for linear Jordan algebras (cf. [12]) and thus the study of minimal ideals in Jordan systems requires different techniques.

For linear Jordan algebras the fact that minimal ideals are either simple or trivial is obtained by Medvedev [12, p. 933] and Skosyrskii [14, Cor. 3.1]. 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 these papers, Nam and McCrimmon study in [13] minimal ideals in quadratic Jordan algebras. In this more general setting, they show that these ideals are either D-simple or trivial. In [5], Block shows that D-simple algebras can be described in terms of simple algebras under the additional assumption of having a minimal ideal. However, nothing is known without this additional condition, so that Nam and McCrimmon leave open the general question on the simplicity of nontrivial minimal ideals. With a different approach, mainly based on the structure theory, in [4] it is shown that the heart of a nondegenerate Jordan (quadratic) algebra, triple system, or pair is simple when nonzero.

1 Partly supported by the Ministerio de Educación y Ciencia and Fondos FEDER, MTM2004-06580-C02-01, and by the Plan de Investigación del Principado de Asturias, FICYT IB05-017.

In this paper we settle the question posed in [13] for quadratic Jordan algebras, obtaining also analogous results for pairs and triple systems, which were unknown even in the linear setting.

Indeed, we focus first on triple systems, and derive from that the corresponding results for algebras and pairs, which gives the shortest approach. However, this generates some combinatorial difficulties due, among other things, to the fact that, in a Jordan triple system, the cube of an ideal need not be an ideal. This is particularly evident in the first section of the paper, devoted to study the multiplication by the elements of a radical ideal, so that a reader only interested in algebras could skip some technicalities. In the second section we obtain the main results of the paper. We first show that minimal ideals are always hearts of certain systems, so that the problem reduces to the degenerate case by using [4]. The proof that the heart of a degenerate Jordan system is trivial is partly inspired by the arguments given by Medvedev in [12]. A crucial fact is the local nilpotency of the McCrimmon radical of a Jordan system, which is a consequence of a result on Lie sandwiches due to Kostrikin and Zelmanov [7].

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 [6, 8, 10, 11] for basic results, notation, and terminology, though we will stress some notions. The identities JPx listed in [8] will be quoted with their original numbering without explicit reference to [8].

—Given a Jordan algebra J, its products will be denoted by 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 have products Q_{x}y ∈ V^{σ}, for any x ∈ V^{σ}, y ∈ V^{−σ}, σ = ±, with linearizations Q_{x,z}y = {x, y, z} = D_{x,y}z.

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

A Jordan algebra or triple system J is said to be trivial if its cube (J^{3} = U_{J}J or P_{J}J, respectively) is zero. A Jordan pair V is called trivial if its cube V^{3} = (Q_{V^{+}}V^{−}, Q_{V^{−}}V^{+}) is zero.

0.2 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^{+} [8, 1.13, 1.14]

0.3 The heart Heart(S) of a Jordan system S is the intersection of all nonzero ideals of S.

1. Multiplication by elements of some degenerate ideals

The algebra analogues of the next two results are immediate consequences of the fact that the cube of an ideal is an ideal, which fails to hold for Jordan triple systems.

1.1 Lemma. P_{J}{I, I, I^{3}} ⊆ I^{3} for any ideal I of a Jordan triple system J.

Proof: Let x ∈ J, a, b, c, d ∈ I. Then u := {b, a, x}, v := P_{x}P_{c}d ∈ I, and
P_{x}{a, b, P_{c}d} = P_{x}L_{a,b}P_{c}d =_{by JP12} (P_{x,{b,a,x}} − L_{b,a}P_{x}) P_{c}d = P_{x,u}P_{c}d + L_{b,a}v
= P_{u,x}P_{c}d + {b, a, v} =_{by JP9} (L_{u,c}L_{x,c} − L_{u,P_{c}x}) d + {b, a, v}
= {u, c, {x, c, d}} − {u, P_{c}x, d} + {b, a, v} ∈ {I, I, I} ⊆ I^{3},

which proves our assertion since I^{3} is spanned by all P_{c}d with c, d ∈ I.

1.2 Lemma. Any minimal ideal I of a Jordan triple system J is either trivial or idempotent (satisfies I^{3} = I).

Proof: Let us suppose that, on the contrary, I^{3} ≠ 0, I. Recall that if H, K are semiideals of J, then so is P_{H}K [9, 6.2(a)], with ideal hull P_{H}K +P_{J}P_{H}K [9, 6.2(b)]. Thus I^{3} is a nonzero semiideal with nonzero hull I^{3} + P_{J}I^{3} ⊆ I, so that minimality of I yields

I = I^{3} + K, where K := P_{J}I^{3} is a semiideal such that K ⊆ I, K ⊈ I^{3}. (1)

If the semiideal P_{I}I^{3} is nonzero, then I = P_{I}I^{3} + P_{J}P_{I}I^{3} as above, so that semiidealness of I^{3} would lead to I ⊆ P_{I}I + P_{J}P_{J}I^{3} ⊆ I^{3} ⊆ I, contrary to our assumption. Therefore P_{I}I^{3} = 0, hence I^{3} = P_{I}I = P_{I}(I^{3} + K) = 0 + P_{I}K using (1). On the other hand, P_{I}P_{J}K = P_{I}P_{J}P_{J}I^{3} ⊆ P_{I}I^{3} again by semiidealness of I^{3}. Thus

P_{I}I^{3} = P_{I}P_{J}K = 0, I^{3} = P_{I}K,  (2)

so that K = P_{J}I^{3} =_{by (2)} P_{J}P_{I}K =_{by (1)} P_{J}P_{I^{3}+K}K. But

— P_{I^{3},K}K = {I^{3}, K, K} ⊆_{by (1)} {I^{3}, I, I},

— P_{I^{3}}K ⊆_{by JP3} P_{I}P_{I}P_{I}K + P_{I^{3},I^{3}}K ⊆_{by (2)} 0 + {I^{3}, K, I^{3}} ⊆_{by (1)} {I^{3}, I, I},

— P_{K}K = P_{P_{J} I^{3}}K ⊆_{by JP3} P_{J}P_{I^{3}}P_{J}K + P_{K,K}K ⊆ P_{J}P_{I}P_{J}K + {K, K, K} ⊆_{by (2)} 0 + {K, K, K} ⊆ {J, J, K} = L_{J,J}K ⊆ K by semiidealness of K, and thus K = P_{J}P_{I^{3}+K}K ⊆ P_{J}({I^{3}, I, I} + K) = P_{J}{I^{3}, I, I} + P_{J}P_{J}I^{3} ⊆ I^{3} by (1.1) and semiidealness of I^{3}, which contradicts (1).

We study now how the elements of certain ideals interact by outer multiplication with the whole multiplication algebra of the system. Given an ideal I of a Jordan triple system J, M(I) denotes the unital subalgebra of End(J) generated by L_{a,b}, P_{c} with a, b, c ∈ I, whereas L_{I,I}, L_{I,J}, L_{J,I}, L_{J,J}, P_{J}, P_{I}, P_{I,I}, P_{J,I} and their products are the submodules of End(J) spanned by the obvious elements. We will also use T_{I} := P_{I} + L_{I,I} and T_{J} := P_{J} + L_{J,J}.

1.3 Lemma. For any ideal I of a Jordan triple system J,

(i) L_{I,I}L_{J,J} ⊆ L_{J,J}L_{I,I} + L_{I,I} ⊆ L_{J,J}M(I) + M(I), (ii) P_{I}L_{J,J} ⊆ L_{J,J}P_{I} + P_{I} ⊆ L_{J,J}M(I) + M(I), (iii) L_{I,I}P_{J} ⊆ P_{J}L_{I,I} + P_{J} ⊆ P_{J}M(I), (iv) L_{I^{3},J} + L_{J,I^{3}} ⊆ L_{I,I} ⊆ M(I),

(v) P_{I^{3}}P_{J} ⊆ P_{I}P_{I} + L_{I,I} + L_{I,I}L_{I,I} ⊆ M(I), (vi) P_{I^{3}}L_{J,J} ⊆ P_{J}M(I), (vii) P_{I^{3}}P_{I^{3}}L_{J,J} ⊆ M(I), (viii) M(I^{3})T_{J} ⊆ T_{J}M(I) + M(I), (ix) If I^{3} = I, then M(I)T_{J} ⊆ T_{J}M(I) + M(I), and P_{I}P_{I}T_{J} ⊆ M(I). Proof: (i) [L_{J,J}, L_{I,I}] ⊆_{by JP15} L_{{J,J,I},I} + L_{I,{J,J,I}} ⊆ L_{I,I}. (ii) L_{J,J}P_{I} + P_{I}L_{J,J} ⊆_{by JP12} P_{I,{J,J,I}} ⊆ P_{I,I} ⊆ P_{I}. (iii) L_{I,I}P_{J} + P_{J}L_{I,I} ⊆_{by JP12} P_{J,{I,I,J}} ⊆ P_{J}. (iv) This is just a triple system version of [13, 1.11]: for a, b ∈ I,

L_{P_{a}b,J} ⊆_{by JP8} L_{a,{b,a,J}} + L_{P_{a}J,b} ∈ L_{I,I}, L_{J,P_{a}b} ⊆_{by JP7} L_{{J,a,b},a} + L_{b,P_{a}J} ∈ L_{I,I}.

(v) First note that P_{I^{3}}P_{J} is spanned by elements of the form P_{P_{a}b}P_{x}, P_{c,d}P_{x} for a, b ∈ I, x ∈ J, c, d ∈ I^{3}. On the one hand,

P_{P_{a}b}P_{x} =_{by JP3} P_{a}P_{b}P_{a}P_{x} =_{by JP21} P_{a} [P_{{x,a,b}} + P_{P_{x}P_{a}b,b} − P_{x}P_{a}P_{b} − L_{x,a}P_{b}L_{a,x}]
=_{by JP12} P_{a} [P_{{x,a,b}} + P_{P_{x}P_{a}b,b} − P_{x}P_{a}P_{b} − L_{x,a} (P_{b,{x,a,b}} − L_{x,a}P_{b})]
=_{by JP13} P_{a} [P_{{x,a,b}} + P_{P_{x}P_{a}b,b} − P_{x}P_{a}P_{b} − L_{x,a}P_{b,{x,a,b}}
+ (L_{P_{x}a,a} + P_{x}P_{a,a}) P_{b}]
= P_{a} [P_{{x,a,b}} + P_{P_{x}P_{a}b,b} + P_{x}P_{a}P_{b} − L_{x,a}P_{b,{x,a,b}} + L_{P_{x}a,a}P_{b}]
=_{by JP3 and JP4} P_{a}P_{{x,a,b}} + P_{a}P_{P_{x}P_{a}b,b} + P_{P_{a}x}P_{b} − P_{a,P_{a}x}P_{b,{x,a,b}}
+ P_{a,P_{a}P_{x}a}P_{b} ∈ P_{I}P_{I},

and, on the other hand,

P_{c,d}P_{x} =_{by JP9} L_{c,x}L_{d,x} − L_{c,P_{x}d} ∈ L_{I^{3},J}L_{I^{3},J} + L_{I,I} ⊆_{by (iv)} L_{I,I}L_{I,I} + L_{I,I}.

(vi) Linearizing JP12 one gets

L_{a,x}P_{b,y} = P_{b,{a,x,y}} + P_{y,{a,x,b}} − P_{b,y}L_{x,a}.  (1)

If c ∈ I^{3}, x, y ∈ J, then

P_{c}L_{x,y} =_{by JP11} L_{c,x}P_{c,y} − P_{P_{c}x,y}
=_{by (1)} P_{c,{c,x,y}} + P_{y,{c,x,c}} − P_{c,y}L_{x,c} − P_{P_{c}x,y}
⊆_{by (iv)} P_{J} + P_{J}L_{I,I}.

(vii) P_{I^{3}}P_{I^{3}}L_{J,J} ⊆_{by (vi)} P_{I^{3}}P_{J}M(I) ⊆_{by (v)} M(I).

(viii) It is enough to notice that L_{J,J}M(I) + M(I) is invariant under left multiplication by the generators of M(I) by (i) and (ii), whereas P_{J}M(I) + M(I) is invariant under left multiplication by the generators of M(I^{3}) by (iii) and (v).

Finally, (ix) follows from (viii), (v), and (vii).

2. Minimal ideals of Jordan Systems

2.1 Lemma. Any minimal ideal of a Jordan system J (algebra, triple system, or pair) is isomorphic to the heart of a homomorphic image of J.

Proof: Given a minimal ideal I of the Jordan system J, let us consider

L := { ideals L of J such that I ∩ L = 0 },

which is easily seen to be a nonempty inductive set, and let L be a maximal element of L. Then, I ≅ I/I ∩ L ≅ (I + L)/L, which is a nonzero ideal Ĩ of J̃:= J/L, and we only have to prove that Ĩ is the heart of J̃, i.e., that it is contained in all nonzero ideals of J̃.

Any nonzero ideal of J̃ has the form M̃ = M/L, where M is an ideal of J strictly containing L. By maximality of L we get M ∩ I ≠ 0, and hence, M ∩ I = I by minimality of I. Thus, M contains I and L, so Ĩ = (I + L)/L ⊆ M/L = M̃.

2.2 Theorem. The heart of a Jordan triple system J is either simple or trivial.

Proof: We may assume that I := Heart(J) is nonzero. If J is nondegenerate, we have the simplicity of I by [4, 3.6], so we only have to deal with a degenerate J. In this case, the McCrimmon radical Mc(J) ([8, §4]) is a nonzero ideal of J, so that I ⊆ Mc(J) and then I = Mc(I) by [8, 4.13], which is locally nilpotent by [3, 2.4(i); 7]. We will show that I^{3} = 0.

Otherwise, since the nonzero heart of a Jordan system is a minimal ideal, I^{3} = I by (1.2).

We claim that I is spanned by absolute zero divisors of J. Indeed, I = I^{3} gives that I is spanned by elements of the form P_{a}b with a, b ∈ I. But [8, 4.6, 4.7] implies that the span of all absolute zero divisors of J is a nonzero ideal of J then containing I. If we write any b ∈ I as a finite sum of absolute zero divisors of J, b = z_{1} +⋯+z_{n}, we get P_{a}b = P_{a}z_{1} + ⋯ + P_{a}z_{n} = x_{1} + ⋯ + x_{n}, where x_{i} := P_{a}z_{i} ∈ I and it is an absolute zero divisor of J by JP3, for all i = 1, …, n.

Let 0 ≠ x ∈ I be an absolute zero divisor of J (for example, one the x_{i}’s above). Obviously, Φx is an inner ideal of J, so that the ideal of J generated by x is its outer

hull 0 ≠ Id_{J}(x) = Out_{J}(x) = ∑_{n=0}^{∞} \overbrace{T_{J} ⋯ T_{J}}^{n} x, where \overbrace{T_{J} ⋯ T_{J}}^{0} x means Φx ([13, 1.9]

can be easily extended to triple systems by replacing the operators U_{x} by operators P_{x} and transformations B_{x,y} [8, 2.11]). Here, x ∈ I implies Id_{J}(x) ⊆ I, and since a

nonzero ideal of J contains its heart I, we get I = Id_{J}(x) = ∑_{n=0}^{∞} \overbrace{T_{J} ⋯ T_{J}}^{n} x.

On the other hand, since x ∈ I = I^{3} = P_{I}I = P_{I}P_{I}I, it must be x = ∑_{finite} P_{a_{i}}P_{b_{i}}c_{i} for some a_{i}, b_{i}, c_{i} ∈ I. Let F be the finite set of elements of J ap-

pearing in the expression of all the a_{i} ′s, b_{i} ′s, and c_{i} ′s as elements in ∑_{n=0}^{∞} \overbrace{T_{J} ⋯ T_{J}}^{n} x.

Let us consider the subsystem J_{0} of J generated by F ∪ {x}, and its ideal I_{0} := Id_{J_{0}}(x) = Out_{J_{0}}(x), so that a_{i}, b_{i}, c_{i} ∈ I_{0} for all i, and then x ∈ P_{I_{0}}P_{I_{0}}I_{0}.

Let us prove I_{0} = I_{0}^{3}. Indeed, we have
I_{0} = Id_{J_{0}}(x) ⊆ Id_{J_{0}}(P_{I_{0}}P_{I_{0}}I_{0}) = P_{I_{0}}P_{I_{0}}I_{0} + P_{J_{0}}P_{I_{0}}P_{I_{0}}I_{0} ⊆ P_{I_{0}}I_{0} = I_{0}^{3} ⊆ I_{0},

using semi-idealness of P_{I_{0}}P_{I_{0}}I_{0} and P_{I_{0}}I_{0} [9, 6.2] , and idealness of I_{0}.

On the other hand, it is clear that I_{0} = Out_{J_{0}}(x) = M(J_{0})x, where the algebra M(J_{0}) has finite length k by [2, 4.2]. So,

I_{0} = I_{0}^{3} = P_{I_{0}}I_{0} = ⋯ = \overbrace{P_{I_{0}} ⋯ P_{I_{0}}}^{2k+1} I_{0} = \overbrace{P_{I_{0}} ⋯ P_{I_{0}}}^{2k+1} ∑_{i=0}^{k} \overbrace{T_{J_{0}} ⋯ T_{J_{0}}}^{i} x.  (1)

If M_{i} := \overbrace{T_{J_{0}} ⋯ T_{J_{0}}}^{i}, we will prove

\overbrace{P_{I_{0}} ⋯ P_{I_{0}}}^{2i} M_{i} ⊆ M(I_{0})  (2)

by induction on i. The containment is obvious for i = 0, and if we assume it for indexes less than i,

\overbrace{P_{I_{0}} ⋯ P_{I_{0}}}^{2i} M_{i} ⊆ \overbrace{P_{I_{0}} ⋯ P_{I_{0}}}^{2i} M_{i−1}T_{J_{0}} ⊆ P_{I_{0}}P_{I_{0}} \overbrace{P_{I_{0}} ⋯ P_{I_{0}}}^{2(i−1)} M_{i−1}T_{J_{0}}
⊆ P_{I_{0}}P_{I_{0}}M(I_{0})T_{J_{0}}
⊆_{by (1.3)(ix)} P_{I_{0}}P_{I_{0}}(T_{J_{0}}M(I_{0}) + M(I_{0})) ⊆_{by (1.3)(ix)} M(I_{0}).

Putting together (1) and (2), and noticing I_{0} ⊆ I, one gets x ∈ I_{0} ⊆ P_{I_{0}}M(I_{0})x ⊆ P_{I}M(I)x, which contradicts the local nilpotency of I.

The following result is an immediate consequence of (2.1) and (2.2).

2.3 Corollary. A minimal ideal of a Jordan triple system is either simple or trivial.

Our next goal is extending the above result to Jordan algebras and Jordan pairs. The algebra case closes the problem posed in [13, p. 582] and it is almost a direct consequence of the relation between algebras and triple systems, whereas the pair situation requires a little more effort, so we study the two cases separately.

2.4 Corollary. A minimal ideal of a Jordan algebra is either simple or trivial.

Proof: Given a Jordan algebra J and one of its minimal ideals I, we consider a unital hull Ĵ, where I is a minimal ideal too. This has the advantage that Ĵ has the same set of ideals when considered as a Jordan triple system, so that I is a minimal ideal of the Jordan triple system Ĵ, and then, either I^{3} = 0 or I is simple as Jordan triple system by (2.3). In the latter case I is obviously simple as Jordan algebra.

2.5 Corollary. A minimal ideal of a Jordan pair is either simple or trivial.

Proof: To apply the known relations between the ideals of a Jordan pair V and the triple system T (V), one needs to assume that V is semiprime (cf. [1, Sect. 5]). Hence, to get the result as a corollary of the proved property for Jordan triple systems, we have to get rid of the non semiprime case. By (2.1) it is enough to prove that the heart I of a Jordan pair V is either simple or trivial.

If V is not semiprime, it contains some nonzero trivial ideal L, which forces I to be trivial since I ⊆ L.

But if V is semiprime, then T (I) = T (Heart(V)) = Heart(T (V)) by [4, 3.7(ii)], so that T (I) is either simple or trivial by (2.2) (indeed, T (I) is necessarily simple by semiprimeness of T (V) [1, Sect. 5]), which implies the same condition on I.

Acknowledgements: The authors want to thank the referee by his or her valuable comments and suggestions, which have brought substantial improvements to the final version of the paper.

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Keywords: Jordan system, heart, minimal ideal, simplicity