Some Characterizations of VNL Rings

Details Some Characterizations of VNL Rings Some Characterizations of VNL Rings

2008-01-16

arxiv.org/abs/0801.2470v1

Mathematics

Rings and Algebras

22 pages

Scientific paper

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.

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