Strongly torsion free, copure flat and Matlis reflexive modules
The author studies strongly torsion-free, copure flat and Matlis reflexive modules. After preliminaries in section 1, he proves in section 2, for a ring \(R\) of finite Krull dimension and an injective \(R\)-module \(E\), the following results: (i) If an \(R\)-module \(M\) is strongly copure flat then \(E\otimes_R M\) is strongly cotorsion; (ii) If \(M\) is strongly copure-injective then \(\text{Hom}_R (E, M)\) is strongly torsion-free. He proves that an \(R\)-module \(M\) is copure flat if and only if \(\text{Ext}^1_R( M,F) = 0\) for all cotorsion flat modules. For a Cohen-Macaulay local ring \(R\), if \(F\) is a flat cotorsion \(R\)-module and \(M\) is a strongly copure flat \(R\)-module, the author proves that \(\text{Hom}_R( M,F)\) is strongly torsion-free. In section 3, the author studies Matlis reflexive modules over a Cohen-Macaulay ring \((R,m)\) of finite Krull dimension. He proves that the completion of a strongly torsion free Matlis reflexive \(R\)-module is a maximal Cohen-Macaulay module over the completion of \(R\). The author closes the paper with the following theorem: If \(M\) is a strongly copure injective Matlis reflexive \(R\)-module then \(\text{Hom}_R(E(R/m), M)\) is either zero or a maximal Cohen-Macaulay module over the completion of \(R\). Suprisingly, the author seems to be not aware of the reviewer's papers on ``Copure submodules'' [\textit{V. A. Hiremath}, Acta Math. Hung. 44, 3--12 (1984; Zbl 0543.16016) and ``Copure injective modules'', Indian J. Pure Appl. Math. 20, No.~3, 250--259 (1989; Zbl 0668.16012)].
scientific article; zbMATH DE number 2093432
scientific article