On the Lie structure of a prime associative superalgebra

TL;DR

This paper investigates the Lie structure of prime associative superalgebras, establishing conditions under which certain substructures are central or contain specific ideals, extending classical results to the superalgebra context.

Contribution

It extends classical Lie structure results to prime superalgebras, providing new conditions for subalgebras and submodules to be central or contain specific ideals.

Findings

If $L$ is a Lie ideal and $W$ a subalgebra with $[W,L] extsubseteq W$, then either $L$ or $W$ is central.

If $V$ is a submodule with $[V,L] extsubseteq V$, then either $V$ or $L$ is central, or there exists a nonzero ideal $M$ with $[M,A] extsubseteq V$.

Results generalize classical Lie structure theorems to the setting of prime superalgebras.

Abstract

In this paper some results on the Lie structure of prime superalgebras are discussed. We prove that, with the exception of some special cases, for a prime superalgebra, $A$, over a ring of scalars $\Phi$ with $1/2\in \Phi$, if $L$ is a Lie ideal of $A$ and $W$ is a subalgebra of $A$ such that $[W, L]\subseteq W$, then either $L\subseteq Z$ or $W\subseteq Z$. Likewise, if $V$ is a submodule of $A$ and $[V, L]\subseteq V$, then either $V\subseteq Z$ or $L\subseteq Z$ or there exists an ideal of $A$, $M$, such that $0\not= [M,A]\subseteq V$. This work extends to prime superalgebras some results of I. N. Herstein, C. Lanski and S. Montgomery on prime algebras.