, Volume 3, Issue 2, pp 273-281

Generalizations of coatomic modules

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Abstract

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ LM| L is a δ-small submodule of M} = Re jm(℘)=∩{ NM: M/N∈℘. We call M δ-coatomic module whenever NM and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R R(RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕ i=1 n Mi is δ-coatomic if and only if each M i (i=1,…, n) is δ-coatomic.