A criterion for rings which are locally valuation rings

Abstract

Abstract. 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 injective.

Modules with respect to cyclic purity

“…There are several generalizations of the notion of purity. Among them, the notion of cyclic purity has been extensively studied by many authors (see, for example, [3], [7], [8], [13], [17]). …”

“…Recall that a left R-module N is CP-injective [17], [7] if for every cyclically pure exact sequence 0 → A → B → C → 0 of left R-modules, the sequence 0 → Hom(C, N ) → Hom(B, N ) → Hom(A, N ) → 0 is exact. A left R-module M is called CP-projective [8] if for every cyclically pure exact sequence 0 → A → B → C → 0 of left R-modules, the sequence 0 → Hom(M, A) → Hom(M, B) → Hom(M, C) → 0 is exact. Clearly, every CP-injective (resp.…”

“…He proved that a Noetherian ring R is pure in every module in which it is cyclically pure if and only if R has small cofinite irreducibles. In [17], Melkersson provided some characterizations for a finitely generated module M over a Noetherian local ring which is pure in every cyclically pure extension of M. In [8], Divaani-Aazar, Esmkhani and Tousi characterized locally valuation rings using the coincidences of cyclic purity and purity. In the present paper, we will study the relation between cyclic purity and purity using a different approach.…”