Some criteria of cyclically pure injective modules

• Published in: Journal of algebra Vol. 304; no. 1; pp. 367 - 381
• Main Authors: Divaani-Aazar, Kamran, Esmkhani, Mohammad Ali, Tousi, Massoud
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-10-2006
• Subjects: Cocyclic modules; Cyclic exact sequences; Cyclically pure injective modules; Quasi-complete rings; Small cofinite irreducibles; Cyclic exact sequences; Cyclically pure injective modules; Cocyclic modules; Small cofinite irreducibles; Quasi-complete rings
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2005.10.023

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