Centralizers of Semiprime Inverse Semirings

Abstract

Let $S$ be a $2$-torsion free semiprime inverse semiring such that all elements of the form $x+x'$ are in the center of $S$. We prove that any additive mapping $F\colon S\to S$ satisfying the condition $2F(xyx)+F(x)yx'+x'yF(x)=0$ is a centralizer.