N-chained semigroups and -perspective modules and rings

Abstract

We introduce and study a new class of modules and rings we call $n/2$-perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small ($\le 2$), we recover existing class of modules and rings: (endo)abelian, strongly IC, and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.

Keywords:

• Associate idempotents
• perspective modules and rings
• reflexive inverses
• (special) clean elements
• (strongly) regular elements and rings
• semigroups

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A09
• 16D70
• 16E50
• 16U99
• 20M99

Acknowledgments

This article benefited greatly, in the final steps of its redaction, of the accurate and relevant comments of Prof. Pace P. Nielsen. It happened (as is not so uncommon in Mathematics) that he and his co-author D. Khuruna had been working at the same time but independently to the present author on a close subject [29], thus sometimes proving the same new results. In particular, the equivalences $(2)\iff (3)$ in [29, Theorems 3.3 and 3.4] is $(3)\iff (4)$ in Theorem 2.5. Also $(1)\iff (2)$ in [29, Theorem 3.13] is Corollary 4.10 and (1) in [29, Proposition 3.19] is Proposition 4.20.

Additional information

Funding

This research has been conducted as part of the project Labex MME-DII [ANR11-LBX-0023-01]