Volume 26, Issue 4
On $π$-Regularity of General Rings

Commun. Math. Res., 26 (2010), pp. 313-320.

Published online: 2021-05

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• Abstract

A general ring means an associative ring with or without identity. An idempotent $e$ in a general ring $I$ is called left (right) semicentral if for every $x ∈ I$, $xe = exe (ex = exe)$. And $I$ is called semiabelian if every idempotent in $I$ is left or right semicentral. It is proved that a semiabelian general ring $I$ is $π$-regular if and only if the set $N(I)$ of nilpotent elements in $I$ is an ideal of $I$ and $I/N(I)$ is regular. It follows that if $I$ is a semiabelian general ring and $K$ is an ideal of $I$, then $I$ is $π$-regular if and only if both $K$ and $I/K$ are $π$-regular. Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring. These generalize several known results on the relevant subject. Furthermore, we give a characterization of a semiabelian GVNL-ring.

• Keywords

semiabelian ring, $π$-regular ring, GVNL-ring, exchange ring.

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@Article{CMR-26-313, author = {Chen , Weixing and Cui , Shuying}, title = {On $π$-Regularity of General Rings}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {26}, number = {4}, pages = {313--320}, abstract = {

A general ring means an associative ring with or without identity. An idempotent $e$ in a general ring $I$ is called left (right) semicentral if for every $x ∈ I$, $xe = exe (ex = exe)$. And $I$ is called semiabelian if every idempotent in $I$ is left or right semicentral. It is proved that a semiabelian general ring $I$ is $π$-regular if and only if the set $N(I)$ of nilpotent elements in $I$ is an ideal of $I$ and $I/N(I)$ is regular. It follows that if $I$ is a semiabelian general ring and $K$ is an ideal of $I$, then $I$ is $π$-regular if and only if both $K$ and $I/K$ are $π$-regular. Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring. These generalize several known results on the relevant subject. Furthermore, we give a characterization of a semiabelian GVNL-ring.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19128.html} }
TY - JOUR T1 - On $π$-Regularity of General Rings AU - Chen , Weixing AU - Cui , Shuying JO - Communications in Mathematical Research VL - 4 SP - 313 EP - 320 PY - 2021 DA - 2021/05 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19128.html KW - semiabelian ring, $π$-regular ring, GVNL-ring, exchange ring. AB -

A general ring means an associative ring with or without identity. An idempotent $e$ in a general ring $I$ is called left (right) semicentral if for every $x ∈ I$, $xe = exe (ex = exe)$. And $I$ is called semiabelian if every idempotent in $I$ is left or right semicentral. It is proved that a semiabelian general ring $I$ is $π$-regular if and only if the set $N(I)$ of nilpotent elements in $I$ is an ideal of $I$ and $I/N(I)$ is regular. It follows that if $I$ is a semiabelian general ring and $K$ is an ideal of $I$, then $I$ is $π$-regular if and only if both $K$ and $I/K$ are $π$-regular. Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring. These generalize several known results on the relevant subject. Furthermore, we give a characterization of a semiabelian GVNL-ring.

WeixingChen & ShuyingCui. (2021). On $π$-Regularity of General Rings. Communications in Mathematical Research . 26 (4). 313-320. doi:
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