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A criterion for rings which are locally valuation rings

Kamran Divaani-Aazar;  Mohammad Ali Esmkhani;  Massoud Tousi — 2009

Colloquium Mathematicae

Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain R is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semisimple if and only if every R-module is cyclically pure...

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi;  Kamran Divaani-Aazar;  Massoud Tousi — 2022

Czechoslovak Mathematical Journal

Let R be a local ring and C a semidualizing module of R . We investigate the behavior of certain classes of generalized Cohen-Macaulay R -modules under the Foxby equivalence between the Auslander and Bass classes with respect to C . In particular, we show that generalized Cohen-Macaulay R -modules are invariant under this equivalence and if M is a finitely generated R -module in the Auslander class with respect to C such that C ⊗ R M is surjective Buchsbaum, then M is also surjective Buchsbaum.

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