Perspectivity and von Neumann regularity

Abstract

We investigate connections between von Neumann regularity of endomorphisms and perspectivity of direct summands in modules. This leads to a new classification of those rings whose regular elements are strongly regular, which turn out to be exactly the rings R whose idempotents are central modulo the Jacobson radical J(R). An important component of our work is an investigation of the left and right associate relations on idempotents, as well as chains of these relations. As applications we give new characterizations of strongly regular elements and of idempotents that are central modulo the Jacobson radical. We also introduce a new class of regular elements that we call pc-regular elements, related to perspectivity in complement summands. These pc-regular elements are exactly the special clean elements. Generalizing the well-known fact that unit-regular rings are special clean, we then show that the unit-regular elements of any regular ring satisfying general comparability are special clean. Consequently, unit-regular endomorphisms of quasi-continuous modules are special clean, answering, in the positive, a conjecture of Lam.

Keywords:

• Perspective (in complements)
• special clean elements and rings
• (strongly) IC rings
• strongly regular elements and rings
• unit-regular elements and rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D70
• Secondary: 16E50
• 16U60
• 16U99

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Acknowledgments

We thank T. Y. Lam for sharing with us his notes on complement perspectivity, and raising the conjecture that led to Theorem 4.11, as well as suggesting conditions (5), (6), and (7) in Theorem 3.13. We also thank Janez Šter for remarks on early versions of these results. Finally, we thank the anonymous referee for a thorough report.