A note on the group structure of unit regular ring elements

Normal Subgroups of the General Linear Groups Over Von Neumann Regular Rings

“…Note that von Neumann regular rings A satisfying sr(A) < 1 are known as unit regular rings, see [7,8,9,11,12,15]. is norr alized by all permutation matrices, we can assume that (i,j) = (l,n).…”

On finite regular rings

“…In this section we shall give necessary and sufficient conditions for the matrix ring R 2X2 to be finite regular or unit regular. We shall need some preliminary results dealing with 2x2 matrices over a regular ring R. We begin by noting that (5.8) and (5.9) are, with aid of regularity, equivalent to But the latter is locally equivalent to a being unit regular [12], which is excluded by assumption. The structure of (5.10) should be compared with the representation R -aR 4-u(a°), u a unit which is equivalent to a being unit regular [12].…”

Rings with internal cancellation

“…These characterizations are to a large extent motivated by [24,Theorem 2B(14)]; in fact, (2) ⇒ (1) below can be gleaned from this reference.…”

“…The first one is the following fact about principal right ideals generated by unit-regular elements in an arbitrary ring R; see [24,Theorem 2B(14)]. Lemma 3.3.…”

“…The converse does not hold in general. The main result in [20] is the following; see also [22,Theorem 1] and [24,Theorem 2B]. Theorem 1.3.…”