Endomorphism rings of nondegenerate modules

Abstract:

Let ${}_RM$ be a left $R$-module whose Morita context is nondegenerate, $S = {\text {End}}({}_RM)$, and $N = \operatorname {Hom} ({}_RM,R)$. If ${}_RM$ is also nonsingular, then the main results of Khuri (Proc. Amer. Math. Soc. 96 (1986), 553-559) are the following: (1) $S$ is left (right) strongly modular if and only if any element of $S$ which has zero kernel in ${}_RM({N_R})$ has essential image in ${}_RM({N_R})$; (2) $S$ is a left (right) Utumi ring if and only if every submodule ${}_RU$ of ${}_RM\;(U_R^{\ast }\;\;{\text {of}} {N_R})$ such that ${U^ \bot } = 0\;({}^ \bot {U^{\ast }} = 0)$ is essential in ${}_RM({N_R})$. In this paper, we show that the same results hold in any nondegenerate Morita context without ${}_RM$ being nonsingular and that $S$ is right nonsingular if and only if ${N_R}$ is nonsingular.