A note on the group structure of unit regular ring elements

TLDR

In ring theory, a ring is regular if for every element a there exists x with axa=a, strongly regular if also axa=a, and unit regular if a has a unit u with aua=a; these notions are defined locally and finite rings satisfy a h=1 ⇒ h a=1. The note aims to demonstrate that a ring element a∈R is unit regular locally if and only if there exists a unit u∈R and a group G⊆R with a∈uG. This is shown by applying the basic theorem for group members in a ring to the element a. Hence unit regular rings are, as it were, locally a rotated version of strongly regular rings.

Abstract

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