On finite regular rings

Schur’s theorem and the Drazin inverse

“…The proof of [7] was based on the fact that Schur's result for matrices over iϊ-rings is really equivalent to the corresponding result for matrices over simple artinian rings (which may be taken to be division rings). Since artinian rings with unity are noetherian [8], p. 69, it follows that artinian rings with unity are strongly-π-regular, so that our local result extends the Schur theorem for artinian rings as proven in [7].…”

A note on the group structure of unit regular ring elements

“…2* Main results* We begin with several preliminary results which will be used in our main theorem. The equivalence of (ii) and (iii) is well-known since ααΓ ^ 66" <==> aa~ ~ 66" and 1 -aa~ ~ 1 -66", while aa~ ~ 66" <=> α~α ^ 6~6, [11]. We note in passing that the statement R/aR = R/bR is clearly equivalent to the statement "aR and bR have all direct summands isomorphic.…”

“…The second equality follows similarly. Before examining the subgroup H e , let us first prove a global conjecture made in [11]. We start with LEMMA 4.…”

“…We remark that when R is a finite regular ring [11] we may shorten this to (3.9) H e = {eue: eueve = e; u, v units in R] .…”

On the Drazin index of regular elements

“…(1). ( (1), (B) shows ∈ 2 R and therefore ∈ R 2 (see [4]), or (C ) implies ∈ R 2 and therefore ∈ 2 R. In either case, # exists.…”