On $π$-Regularity of General Rings (Communications in Mathematical Research, 2021)
A general ring means an associative ring with or without identity. Anidempotent $e$ 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 leftor right semicentral. It is proved that a semiabelian general ring $I$ is $π$-regular ifand only if the set $N(I)$ of nilpotent elements in $I$ is an ideal of $I$ and $I/N(I)$ isregular. 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 weprove that every semiabelian GVNL-ring is an SGVNL-ring. These generalize severalknown results on the relevant subject. Furthermore, we give a characterization of asemiabelian GVNL-ring.