A note on inverse limits of injective modules

Abstract

Let $R$ be a ring and $M$ a left $R$-module. We prove that $M$ is an $\omega_1$-indexed inverse limit of injective modules with split epimorphic connecting maps, and $M$ is also an $\omega$-indexed inverse limit of a stationary tower defined by a split monic endomorphism of one injective module. These constructions answer two questions of Bergman. We then study the class of direct summands of $\omega$-indexed inverse limit of injective modules with epimorphic connecting maps. Over a commutative Noetherian ring $R$, they are precisely the Enochs cotorsion modules if and only if $\operatorname{gldim}R\leq1$ and $\operatorname{Spec}R$ is countable.