Growing Hearts in Associative Systems

Abstract

<jats:p>We show that associative systems with a sufficiently good module structure imbed in a primitive system with simple primitive heart, spanned by the original system and the heart, so extending the results of J. Pure Appl. Algebra 181 (2003) 131–139 to systems over more general rings of scalars. We also study associative systems with involution.</jats:p>

Prime Quotients of Jordan Systems and Lie Algebras

“…For any x, y ∈ L, the equalities xy + yx = (x + y) 2 − x 2 − y 2 and xy + yx = −(x − y) 2 + x 2 + y 2 allow us to use (1) and (3) respectively, and M will be an ideal of L not contained in P , by primeness of P [the elements x, y and x + y or x − y are not contained in P , so their Lie products with L cannot be contained in P by semiprimeness of P , as in (5); thus M is the product of three ideals, each one not contained in P ].…”

Imbedding Lie Algebras in Strongly Prime Algebras

“…The first section is devoted to study centralizers and absolute zero divisors of subsets of endomorphisms acting densely on vector spaces. This sets appear naturally in the constructions of [2] and [3] for associative algebras. The results of Section 1 will be applied to give in the next section a modified version of the main construction of [2] for associative algebras over fields, specifically prepared to deal comfortably with nondegeneracy when applying it to Lie algebras.…”

“…This was aimed at building counterexamples linked to some problems with Jordan systems [1]. Recently, in [3], the results obtained in [2] were improved in two senses: on the one hand, associative systems over more general rings of scalars were considered, and, on the other hand, analogues for systems with involution were also obtained.…”

“…The results of Section 1 will be applied to give in the next section a modified version of the main construction of [2] for associative algebras over fields, specifically prepared to deal comfortably with nondegeneracy when applying it to Lie algebras. From that we will obtain a Lie analogue of the main theorem of [3] for Lie algebras over fields. In the final third section, we will obtain our main theorems, which match the associative analogues of [3]: we will find a necessary condition on the underlying module of a Lie algebra L to find a prime Lie algebra where L imbeds and see that the condition is also sufficient.…”

“…From that we will obtain a Lie analogue of the main theorem of [3] for Lie algebras over fields. In the final third section, we will obtain our main theorems, which match the associative analogues of [3]: we will find a necessary condition on the underlying module of a Lie algebra L to find a prime Lie algebra where L imbeds and see that the condition is also sufficient. Indeed, it guaranties that L imbeds in a strongly prime Lie algebra with simple nondegenerate heart, spanned as a direct sum by the isomorphic image of L and the heart.…”