Some criteria of cyclically pure injective modules

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 HomR(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,m) an R-module D is cyclically pure injective if and only if there is a family {Cλ}λ∈Λ of cocyclic modules such that D is isomorphic to a direct summand of ∏λ∈ΛCλ. Finally, we show that over a complete local ring every finitely generated module which has small cofinite irreducibles is cyclically pure injective.