  Let R be a ring (not necessarily commutative). A left R-module is said to be
cotorsion if Ext 1 R (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 Ext 1 R (R/A, M) = 0 for each pure ideal A of R
(Baer's criterion).

Rings and Algebras;Commutative Algebra