Another generalization of a theorem of A. Kertész. (English) Zbl 0594.16013
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 \(B^{k+1}=B^{h+1}\).
Reviewer: F.Loonstra
MSC:
16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
16Dxx | Modules, bimodules and ideals in associative algebras |
References:
[1] | A. Kertész, Zur Frage der Spartbarkeit von Ringen,Bull. Acad. Polonaise Sci., Serie math. ast. phys.,12 (1964), 91–93. |
[2] | H. Tominaga, Ons-unital rings,Math. J. Okayama Univ.,18 (1976), 117–134. · Zbl 0335.16020 |
[3] | H. Tominaga, A generalization of a theorem of A. Kertész,Acta Math. Acad. Sci. Hungar.,33 (1979), 323. · doi:10.1007/BF01902567 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.