On the splitting property of rings with restricted class of injectivity domains

Abstract

There has been a growing research interest in exploring rings whose modules are either injective or far from being injective to a specified degree. As is commonly observed in each of such scenarios, rings can be broken as a direct sum of a semisimple Artinian ring and an indecomposable ring having zero or (essential) homogeneous right socle. Although this phenomenon has become increasingly expected and less surprising with every new study, proving it has never become easier, and in most cases, it has been obtained as a result of considerable efforts. This work aims to bring such studies to a common denominator and tries to explain why this phenomenon occurs each time we deal with the same type of problem.

KEYWORDS:

• Hereditary pretorsion class
• indecomposable ring
• injectivity domain
• left exact preradical
• socle

2020MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D50
• Secondary: 16S90
• 16N60

Acknowledgments

The author gratefully acknowledges the support he has received from Turkish Scientific Research Council (TUBITAK) with Grant No. 117F084.

Author contributions

The author is responsible for the entire study.

Disclosure statement

The author has no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Ethical approval

It is not applicable.

Additional information

Funding

It is not applicable.