ON VNL-RINGS AND n-VNL-RINGS

Abstract

A ring R is called a VNL-ring if a or 1−a is regular for every a ∈ R. We call a ring R a right n-VNL-ring if whenever a1R + a2R + · · · + anR = R for some elements a1, a2, · · · , an of R, then there exists at least one element ai (von Neumann) regular. It is proven that there exists a right 2-VNL-ring but not right 3-VNL, which gives a negative answer to a question raised by Chen and Tong in 2006. We prove that R is regular iff the n × n matrix ring over R is a VNL-ring for some n ≥ 2. It is also proven that a ring R is a division ring iff R is semilocal and the 2 × 2 upper triangular matrix ring over R is a VNL-ring.

Keywords

• division ring, local ring, n-VNL-ring, regular ring, VNL-ring