Some Characterizations of VNL Rings

Abstract

A ring R is said to be von Newmann local (VNL) if for any a ∈ R, either a or 1 −a is (von Neumann) regular. The class of VNL rings lies properly between exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without an infinite set of orthogonal idempotents; and also the VNL rings having a 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 ₁, a ₂) ∈ R ², one of the a _i 's 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.

Key Words:

• Exchange rings
• Semiperfect rings
• VNL rings

2000 Mathematics Subject Classification:

• Primary 16E50
• Secondary 16L99

ACKNOWLEDGMENT

We are highly grateful to the referee for pointing out many errors and also suggesting the inclusion of proofs of various statements. The research of the first author is supported by CSIR, India, and will form a part of her PhD Thesis.

Notes

A ring is called abelian if all its idempotents are central.

Communicated by E. Puczylowski.