Equations Related to Superderivations on Prime Superalgebras

Abstract

In this paper we investigate equations related to superderivations on prime superalgebras. We prove the following result. Let $D=D_0+D_1$ be a nonzero superderivation on a prime associative superalgebra $\mathscr{A}$ satisfying the relations $D_i(x)[D_i(x),x]_s=0$, $[D_i(x),x]_sD_i(x)=0$ for all $x\in \mathscr{A}$, $i=0,1$. Then one of the following is true: (a) $\mathscr{A}_1=0$ and $D(\mathscr{A}_0) \subseteq Z(\mathscr{A})$ or (b) $ D(\mathscr{A}_0)=0$ and $\mathscr{A}$ is commutative or (c) $D^2=0$. The research is a generalization of the results in [10] and [4] by using the theory of superalgebras.