A note on the group structure of unit regular ring elements

1* Introduction* It is well-known that [15, 7] a ring R is strongly regular if and only if every aeR is a group member. In this note we shall use the basic theorem for group members in a ring to show locally that a ring element a e R (with unity) is unit regular exactly when there is a unit ue R and a group G in R such that a e uG. Hence unit regular rings are, as it were locally a rotated version of strongly regular rings. We remind the reader that a ring R is called regular if for every aeR, aeaRa; strongly regular if for every aeR, aeaR, and unit regular if for every aeR, there is a unit u e R such that ana = a [3]. Similar definitions hold locally. A ring with unity is called finite if ah = 1 implies ha — 1. Any solution a~ to axa = a is called an inner or 1-inverse of [1], while any solution a to axa = a and xax = x is called a reflexive or 1-2 inverse of a. For idempotents e and / in R, e ~ / denotes the equivalence in