Properties of endomorphism rings of modules and their duals

Khuri, Soumaya Makdissi

doi:10.1090/S0002-9939-1986-0826480-8

Properties of endomorphism rings of modules and their duals
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by Soumaya Makdissi Khuri PDF

Proc. Amer. Math. Soc. 96 (1986), 553-559 Request permission

Abstract:

Let $_RM$ be a nonsingular left $R$-module whose Morita context is nondegenerate, let $B = \operatorname {End}_{R}M$ and let ${M^ * } = \operatorname {Hom}_{R}(M,R)$. We show that $B$ is left (right) strongly modular if and only if any element of $B$ which has zero kernel in $_RM(M_R^ * )$ has essential image in $_RM(M_R^ * )$, and that $B$ is a left (right) Utumi ring if and only if every submodule $_RU{\text {o}}{{\text {f}}_R}M(U_R^ * {\text {of }}M_R^ * )$ such that ${U^ \bot } = 0{(^ \bot }{U^ * } = 0)$ is essential in $_RM(M_R^ * )$.

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