Model-theoretic aspects of Sigma-cotorsion modules

Let R be an associative ring with identity. It is shown that every Σ -cotorsion left R-module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ -cotorsion. It follows that, for countable R, the class of Σ -cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them as the modules M for which Ext1(F,M(א0)) = 0, for every countably presented flat module F . c © 2006 Elsevier B.V. All rights reserved.