Co-Commuting Mappings of Generalized Matrix Algebras

Abstract

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$ and $Z(\mathcal{G})$ be the center of $\mathcal{G}$. Suppose that $F, T : \mathcal{G} → \mathcal{G}$ are two co-commuting $\mathcal{R}$-linear mappings, i.e., $F(x)x = xT(x)$ for all $x ∈ \mathcal{G}$. In this note, we study the question of when co-commuting mappings on $\mathcal{G}$ are proper.