An introduction to q-central idempotents and q-abelian rings

Abstract

A ring idempotent $e\in R$ is said to be quarter-central (or q-central for short) if $eR(1-e)Re=0$. If all idempotents in a ring $R$ are q-central, $R$ is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring $R$ is characterized by the property that $[e,R][e,R]=0$ for all idempotents $e\in R$, where $[x,y]$ denotes the additive commutator $xy-yx$. For any nonzero ring $S$ and any integer $n\ge 3$, we show that the ring ${\mathbb{T}}_n(S)$ of $n\times n$ upper triangular matrices over $S$ is not q-abelian. On the other hand, ${\mathbb{T}}_2(S)$ is q-abelian iff $S$ is abelian (in the classical sense that all idempotents are central in $S$). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings.

2020 Mathematics Subject Classification:

• 16E50
• 16N40
• 16N60
• 16U60
• 16U80

Keywords:

• Exchange rings
• regular and strongly regular elements in rings
• semicentral and q-central idempotents
• semiabelian and q-abelian rings
• upper triangular matrix rings

Acknowledgments

It is the author’s pleasure to heartily thank Professors P. N. Ánh, G. Birkenmeier, D. Khurana, X. Mary and P. Nielsen for their various helpful comments and suggestions on this paper. Professors Ánh and Birkenmeier have further pointed out that some of the results on $2\times 2$ upper triangular matrix rings ${\mathbb{T}}_2(S)$ proved in this paper can be suitably extended to generalized matrix rings of the form $\left(\begin{array}{cc}S& {}_S{M}_T\\ {}_T{N}_S& T\end{array}\right)$ with trivial “pairing maps” $M{\otimes }_{T}N\mapsto 0\in S$ and $N{\otimes }_{S}M\mapsto 0\in T$.

Notes

1 We should point out here that some different names have been used in the recent literature. In [24], such a ring was called “2-abelian”, while in [36], such a ring was said to be “quasi-normal”.

2 In this way, the notion of “$e$ being q-central in $R$” would become meaningful also for any ring $R$ possibly without an identity element. However, in this paper, all rings are assumed to be unital.