SUPERCENTRALIZING AUTOMORPHISMS ON PRIME SUPERALGEBRAS

Abstract

Let $A=A_{0}\oplus A_{1}$ be a noncommutative prime superalgebra over a commutative associative ring $F$ with $\frac{1}{2}\in F$. Let $Z_{s}(A)$ be the supercenter of $A$. If an $Z_{2}$-preserving automorphism $\varphi: A\rightarrow A$ satisfies $[\varphi(x),x]_{s}\in Z_{s}(A)$ for all $x\in A$, then $\varphi=1$, where $1$ denotes the identity map of $A$. Moreover, if $A_{1}\neq 0$, then $A$ is a central order in a quaternion algebra. This gives a version of Mayne's theorem for superalgebras.