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 Φ with 1/2 ∈ Φ, if L is a Lie ideal of A and W is a subalgebra of A such that [. W, L] ⊆ W, then either L ⊆ Z or W ⊆ Z. Likewise, if V is a submodule of A and [. V, L] ⊆ V, then either V ⊆ Z or L ⊆ Z or there exists an ideal of A, M, such that 0 ≠ [. M, A] ⊆ V. This work extends to prime superalgebras some results of I.N. Herstein, C. Lanski and S. Montgomery on prime algebras. © 2014 Elsevier Inc.