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 - <p style="text-align: justify;">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.</p>