On the Lie structure of a prime associative superalgebra

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 $0\ne [M,A]\subseteq V$. This work extends to prime superalgebras some results of I. N. Herstein, C. Lanski and S. Montgomery on prime algebras.