Morita duality for endomorphism rings

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.