Another generalization of a theorem of A. Kertész (Q1076774)

A ring R is left s-unital if \(x\in Rx\) (\(\forall x\in R)\). If F is a finite subset of a left s-unital ring R, then \(\exists e\in R\) with \(x=ex\) (\(\forall x\in F)\). Let A be an ideal of R, B an additive subgroup of R such that \(R=A+B\). Let R/A be left s-unital and \(AB=BA=0\), then \(R=A\oplus B^ 2\) (as a ring direct sum). This result (of Tominaga) has now been generalized as follows: let A be an ideal of R, and B an additive subgroup of R such that \(R=A+B\). If R/A is left s-unital and there exist positive integers \(k>h\) such that \(B^ kA\subseteq B^ k\), \(AB^ h\subseteq B^ h\), and \(B^ kA\cap B^{k+1}=0=AB^ h\cap B^{h+1}\), then R is the (ring-)direct sum \(A\oplus B^{k+1}\). If moreover, R is left s-unital (the right annihilator of R is 0), then \(B^{k+1}=B^{h+1}\).