Decomposition of modules into modules whose annihilators are pseudo-irreducible

ABSTRACT

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 $M={\oplus }_{i=1}^n{N}_i$, 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.

KEYWORDS:

• API-module
• complete comaximal decomposition
• pseudo-irreducible ideal
• von Neumann regular ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C99, 13C13

Acknowledgments

The authors gratefully acknowledge the constructive comments on this paper offered by the anonymous referee. We express our sincere gratitude for his/her review, which helped to improve the quality of the paper significantly.