Modules in which Complements are Summands

Abstract

<p>This thesis studies modules over commutative integral domains with the property that every closed submodule is a direct summand (we denote this property by (C₁)). It is shown that any non-torsion module with property (C₁) is a direct sum of an injective submodule and a finite direct sum of uniform torsion free reduced submodules. This reduces the study of the problem to finite direct sums of uniform torsion free reduced modules and to torsion modules. Then we characterize finite direct sums of uniform torsion free reduced modules over commutative (Prüfer, Noetherian of Krull dimension one, Dedekind) domains which have property (C₁). We also characterize finite direct sums of uniform torsion modules with local endomorphism rings over Noetherian domains which have property (C₁). Finally, we classify all modules with property (C₁) over Dedekind domains.</p>