Simple-Direct Modules Over Formal Matrix Rings

Abstract

In the present paper, we study simple-direct-injective modules and simple-direct-projective modules over a formal matrix ring $$K=\left(\begin{matrix}R&M\\ N&S\end{matrix}\right)$$ , where $$M$$ is an $$(R,S)$$ -bimodule and $$N$$ is a $$(S,R)$$ -bimodule over rings $$R$$ and $$S$$ . We determine necessary and sufficient conditions for a $$K$$ -module to be, respectively, simple-direct-injective or simple-direct-projective. We also give some examples to illustrate and delimit our results.