Almost Abelian rings

Wei, Junchao. "Almost Abelian rings." Communications in Mathematics 21.1 (2013): 15-30. <http://eudml.org/doc/260701>.

@article{Wei2013,
abstract = {A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N(R)$ and $e\in E(R)$, where $E(R)$ and $N(R)$ stand respectively for the set of idempotents and the set of nilpotents of $R$. Some characterizations and properties of such rings are included. It follows that if $R$ is a left almost Abelian ring, then $R$ is $\pi $-regular if and only if $N(R)$ is an ideal of $R$ and $R/N(R)$ is regular. Moreover it is proved that (1) $R$ is an Abelian ring if and only if $R$ is a left almost Abelian left idempotent reflexive ring. (2) $R$ is strongly regular if and only if $R$ is regular and left almost Abelian. (3) A left almost Abelian clean ring is an exchange ring. (4) For a left almost Abelian ring $R$, it is an exchange $(S,2)$ ring if and only if $\mathbb \{Z\}/2\mathbb \{Z\}$ is not a homomorphic image of $R$.},
author = {Wei, Junchao},
journal = {Communications in Mathematics},
keywords = {left almost Abelian rings; $\pi $-regular rings; Abelian rings; $(S,2)$ rings; left almost Abelian rings; -regular rings; -rings; nilpotent elements; idempotents},
language = {eng},
number = {1},
pages = {15-30},
publisher = {University of Ostrava},
title = {Almost Abelian rings},
url = {http://eudml.org/doc/260701},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Wei, Junchao
TI - Almost Abelian rings
JO - Communications in Mathematics
PY - 2013
PB - University of Ostrava
VL - 21
IS - 1
SP - 15
EP - 30
AB - A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N(R)$ and $e\in E(R)$, where $E(R)$ and $N(R)$ stand respectively for the set of idempotents and the set of nilpotents of $R$. Some characterizations and properties of such rings are included. It follows that if $R$ is a left almost Abelian ring, then $R$ is $\pi $-regular if and only if $N(R)$ is an ideal of $R$ and $R/N(R)$ is regular. Moreover it is proved that (1) $R$ is an Abelian ring if and only if $R$ is a left almost Abelian left idempotent reflexive ring. (2) $R$ is strongly regular if and only if $R$ is regular and left almost Abelian. (3) A left almost Abelian clean ring is an exchange ring. (4) For a left almost Abelian ring $R$, it is an exchange $(S,2)$ ring if and only if $\mathbb {Z}/2\mathbb {Z}$ is not a homomorphic image of $R$.
LA - eng
KW - left almost Abelian rings; $\pi $-regular rings; Abelian rings; $(S,2)$ rings; left almost Abelian rings; -regular rings; -rings; nilpotent elements; idempotents
UR - http://eudml.org/doc/260701
ER -