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