Model-theoretic aspects of [formula omitted]-cotorsion modules

Abstract

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 Ext 1 ( F , M ( ℵ 0 ) ) = 0 , for every countably presented flat module F .