QF-3' rings and Morita duality
The main theorem of this paper is the following characterization of QF-3' rings. Theorem 1. For any ring R, the following are equivalent: (a) R is left QF-3'; (b) The double dual functor ( )\({}^{**}\) preserves monomorphisms in R-Mod; (c) If \(i: R\to E\) is the inclusion of R into its injective envelope E in R-Mod, then \(i^{**}\) is a monomorphism. In view of this theorem results of the authors [J. Algebra 82, 546-558 (1983; Zbl 0524.16014)] provide Theorem 2: For any ring R, the following are equivalent: (1) R is a left QF-3' ring and its own maximal left quotient ring; (2) The double dual functor ( )\({}^{**}\) is left exact on R-Mod; (3) If \(0\to R\to^{i}E_ 1\to^{j}E_ 2\) is exact with \(E_ 1\) and \(E_ 2\) injective in R-Mod then \(0\to R\to^{i^{**}}E_ 1^{**}\to^{j^{**}}E_ 2^{**}\) is also exact. These theorems were applied to obtain the following characterization of Morita duality between Grothendieck categories. Theorem 6: Every right adjoint pair of contravariant faithful functors \(D: {\mathfrak a}\rightleftharpoons {\mathfrak a}': D'\) between Grothendieck categories with reflexive generators defines a Morita duality. As a specialization of this theorem the authors have proved Theorem 7: If R and S are rings and \({}_ RU_ S\) is a bimodule with \(S=End(_ RU)\) and \(R=End(U_ S)\) and if \({}_ RU\) and \(U_ S\) are cogenerators, then \({}_ RU\) and \(U_ S\) are injective. This new characterization of Morita duality between categories of modules generalizes \textit{T. Onodera}'s theorem that cogenerators are injective [Arch. Math. 19, 402-410 (1968; Zbl 0165.051)].
scientific article; zbMATH DE number 3935248