  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.
