Decomposition of modules into modules whose annihilators are pseudo-irreducible

摘要

A nonzero module M over a commutative ring R is said to have a complete comaximal decomposition if it can be written in the form where the annihilators of the N-i's are pairwise comaximal pseudo-irreducible ideals of R. In this paper, we show that a complete comaximal decomposition for an R-module is unique if it exists and a ring R is J-Noetherian if and only if every nonzero R-module has a complete comaximal decomposition. Also, we give a topological characterization of (finitely generated) comultiplication R-modules whose submodules have a complete comaximal decomposition and then we show that a commutative ring R is von Neumann regular if and only if every representable R-module can be written as a finite direct sum of homogeneous semi-simple R-modules.