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Proceedings of the American Mathematical Society

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

 

 

Morita duality for endomorphism rings


Authors: Robert W. Miller and Darrell R. Turnidge
Journal: Proc. Amer. Math. Soc. 31 (1972), 91-94
DOI: https://doi.org/10.1090/S0002-9939-1972-0286847-6
MathSciNet review: 0286847
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Abstract: A ring $ R$ is said to have a left Morita duality with a ring $ S$ if there is an additive contravariant equivalence between two categories of left $ R$-modules and right $ S$-modules which include all finitely generated modules in $ _R\mathfrak{M}$ and $ {\mathfrak{M}_S}$ respectively and which are both closed under submodules and homomorphic images. We show that for such a ring $ R$ the endomorphism ring of every finitely generated projective left $ R$-module $ _RP$ has a left Morita duality with the endomorphism ring of a suitably chosen cofinitely generated injective left $ R$-module $ _RQ$. Specialized to injective cogenerator rings and quasi-Frobenius rings our results yield results of R. L. Wagoner and Rosenberg and Zelinsky giving conditions when the endomorphism ring of a finitely generated projective left module over an injective cogenerator (quasi-Frobenius) ring is an injective cogenerator (quasi-Frobenius) ring.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0286847-6
Keywords: Morita duality, endomorphism rings
Article copyright: © Copyright 1972 American Mathematical Society