Some Characterizations of VNL Rings

TL;DR

This paper explores the properties and classifications of VNL rings, a class of rings where for each element, either it or its complement is von Neumann regular, providing characterizations and specific conditions for various types.

Contribution

It offers new characterizations and classifications of VNL rings, including abelian, semiperfect, and triangular matrix rings, expanding understanding of their structure and properties.

Findings

Abelian VNL rings are characterized.

VNL rings without infinite orthogonal idempotents are classified.

Upper triangular matrix rings are VNL only for specific sizes and rings.

Abstract

A ring R is said to be VNL if for any a in R, either a or 1-a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without infinite set of orthogonal idempotents; and also the VNL rings having primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a, b) in R^2, one of the a or b is regular in R. Formal triangular matrix rings that are VNL, are also characterized. As a corollary it is shown that an upper triangular matrix ring T_n(R) is VNL if and only if n=2 or 3 and R is a division ring.