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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Properties of endomorphism rings of modules and their duals


Author: Soumaya Makdissi Khuri
Journal: Proc. Amer. Math. Soc. 96 (1986), 553-559
MSC: Primary 16A08; Secondary 16A65
DOI: https://doi.org/10.1090/S0002-9939-1986-0826480-8
MathSciNet review: 826480
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0826480-8
Keywords: Endomorphism rings, nonsingular modules and rings, nondegenerate Morita contexts, maximal quotient rings, Utumi rings
Article copyright: © Copyright 1986 American Mathematical Society