Modules whose endomorphism rings have isomorphic maximal left and right quotient rings

Abstract:

Let $_RM$ be a left $R$-module such that ${\operatorname {Hom} _R}(M,U) \ne 0$ for any nonzero submodule $U$ of $M$, let $E(M)$ denote the injective hull of $M$, and let $B$ (resp. $A$) denote the ring of $R$-endomorphisms of $M$ (resp. $E(M)$). It is known that if $M$ is nonsingular then $B$ is left nonsingular and $A$ is the maximal left quotient ring of $B$. We give here necessary and sufficient conditions on $M$ for $B$ to be right nonsingular and for $A$ to be the maximal right quotient ring of $B$.