(untitled) (Communications in Mathematical Research, 2021)
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.