Finite Σ-Rickart modules

Abstract

After hereditary rings were defined, as a natural generalization, semi-hereditary rings were introduced. In this article, we study the notion of a finite Σ-Rickart module, as a module theoretic analogue of a right semi-hereditary ring. A module M is called finite Σ-Rickart if every finite direct sum of copies of M is a Rickart module. It is shown that any direct summand and any finite direct sum of copies of a finite Σ-Rickart module are finite Σ-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of semi-hereditary rings: a ring R is right semi-hereditary if and only if every finitely generated submodule of any projective right R-module is projective if and only if every factor module of any injective right R-module is f-injective. Further, we have a characterization of a finite Σ-Rickart module in terms of its endomorphism ring. In addition, we introduce M-coherent modules and provide a characterization of a finite Σ-Rickart module in terms of an M-coherent module. We prove that for any endoregular module M, ${\text{Hom}}_R(M,A)$ is an f-injective right S-module for any module A, where $S={\text{End}}_R(M).$ Examples which delineate the concepts and results are provided.

Keywords:

• Finite Σ-Rickart module
• f-injective module
• M-coherent module
• M-pure epimorphism
• Rickart module
• semi-hereditary ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D40
• 16E60
• 16S50
• Secondary: 16D50
• 16E50

Acknowledgments

The authors express their sincere gratitude to the referee for a prompt and thorough report and are very thankful to Research Institute of Mathematical Sciences, Chungnam National University (CNU-RIMS), Republic of Korea, for the support of this research work. The first author gratefully acknowledges the support of this research work by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT)(2019R1F1A1059883)