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- Title
Commutative rings whose cotorsion modules are pure-injective.
- Authors
Couchot, François
- Abstract
Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if ExtR¹(G,M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for instance, if R is left perfect but not left pure-semisimple then, each left R-module is cotorsion but there exist non-pure-injective left modules. The aim of this paper is to describe the class C of commutative rings R for which each cotorsion R-module is pure-injective. It is easy to see that C contains the class of von Neumann regular rings and the one of pure-semisimple rings. We prove that C is strictly contained in the class of locally pure-semisimple rings. We state that a commutative ring R belongs to C if and only if R verifies one of the following conditions: (1) R is coherent and each pure-essential extension of R-modules is essential; (2) R is coherent and each RD-essential extension of R-modules is essential; (3) any R-module M is pure-injective if and only if ExtR¹ (R/A,M) = 0 for each pure ideal A of R (Baer's criterion).
- Subjects
RING extensions (Algebra); COMMUTATIVE rings; ARTIN rings
- Publication
Palestine Journal of Mathematics, 2016, Vol 5, p81
- ISSN
2219-5688
- Publication type
Article