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On Cofinitely Injective Modules
Author(s) -
Hiremath V. A.
Publication year - 1978
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-17.1.28
Subject(s) - injective module , injective function , mathematics , simple module , divisible group , torsion (gastropod) , module , pure mathematics , dedekind cut , ring (chemistry) , flat module , discrete mathematics , simple (philosophy) , torsion subgroup , chemistry , medicine , philosophy , abelian group , surgery , epistemology , organic chemistry , elementary abelian group
In this paper, we introduce the notion of cofinitely injective modules and study their properties over special types of rings. We define a right R ‐module M to be cofinitely injective if it is injective with respect to short exact sequences of right R ‐modules of the form 0→A→B→C→0, where A is cofinitely generated. We completely characterize cofinitely injective modules over a right conoetherian ring, as precisely those modules M , whose torsion part L ( M ) in the simple torsion theory ( L , L ) contains an essential submodule B = → l i m { B 1 }1 ∈ Iwhere{ B 1 }1 ∈ Iis a directed system of injective submodules of L ( M ) such that every cofinitely generated submodule of M is contained in some B 1 . This leads to a complete characterization of cofinitely injective modules over a Dedekind domain. Over a right artinian ring, the cofinitely injective modules are precisely the injective modules.