Morita duality for endomorphism rings

Miller, Robert W.; Turnidge, Darrell R.

doi:10.1090/S0002-9939-1972-0286847-6

Morita duality for endomorphism rings
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by Robert W. Miller and Darrell R. Turnidge

Proc. Amer. Math. Soc. 31 (1972), 91-94

DOI: https://doi.org/10.1090/S0002-9939-1972-0286847-6

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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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Bibliographic Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 91-94
DOI: https://doi.org/10.1090/S0002-9939-1972-0286847-6
MathSciNet review: 0286847