On the Lie structure of a prime associative superalgebra

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.