Some criteria of cyclically pure injective modules

Abstract

The structure of cyclically pure injective modules over a commutative ring $R$ is investigated and several characterizations for them are presented. In particular, we prove that a module $D$ is cyclically pure injective if and only if $D$ is isomorphic to a direct summand of a module of the form $\Hom_R(L,E)$ where $L$ is the direct sum of a family of finitely presented cyclic modules and $E$ is an injective module. Also, we prove that over a quasi-complete Noetherian ring $(R,\fm)$ an $R$-module $D$ is cyclically pure injective if and only if there is a family $\{C_{\lambda }{\}}_{\lambda \in \Lambda }$ of cocyclic modules such that $D$ is isomorphic to a direct summand of ${\Pi }_{\lambda \in \Lambda }{C}_{\lambda }$. Finally, we show that over a complete local ring every finitely generated module which has small cofinite irreducibles is cyclically pure injective.

18 pages, to appear in Journal of Algebra