Co-commuting Mappings of Generalized Matrix Algebras

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 $mathcal{R}$-linearmappings, i.e., $F(x)x = xT(x)$ for all $x ∈ mathcal{G}$. In this note, we study the question ofwhen co-commuting mappings on $mathcal{G}$ are proper.