Finite $\Sigma$-Rickart modules
Abstract
In this article, we study the notion of a finite $\Sigma$-Rickart module, as a module theoretic analogue of a right semi-hereditary ring. A module $M$ is called \emph{finite $\Sigma$-Rickart} if every finite direct sum of copies of $M$ is a Rickart module. It is shown that any direct summand and any direct sum of copies of a finite $\Sigma$-Rickart module are finite $\Sigma$-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of semi-hereditary rings. Also, we have a characterization of a finite $\Sigma$-Rickart module in terms of its endomorphism ring. In addition, we introduce $M$-coherent modules and provide a characterization of finite $\Sigma$-Rickart modules in terms of $M$-coherent modules. At the end, we study when $\Sigma$-Rickart modules and finite $\Sigma$-Rickart modules coincide. Examples which delineate the concepts and results are provided.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2021
- DOI:
- 10.48550/arXiv.2102.01014
- arXiv:
- arXiv:2102.01014
- Bibcode:
- 2021arXiv210201014L
- Keywords:
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- Mathematics - Rings and Algebras;
- 16D40;
- 16D50;
- 16E50;
- 16E60;
- 16S50
- E-Print:
- This is a modified and extended version of the version submitted for publication