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Proceedings of the American Mathematical Society

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



Endomorphism rings of nondegenerate modules

Author: Zheng Ping Zhou
Journal: Proc. Amer. Math. Soc. 120 (1994), 85-88
MSC: Primary 16D90; Secondary 16S50
MathSciNet review: 1161402
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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.

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Article copyright: © Copyright 1994 American Mathematical Society