On π-regularity of General Rings

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.