Extensions of strongly π-regular rings

An ideal I of a ring R is strongly π -regular if for any x ∈ I there exist n ∈ ℕ and y ∈ I such that xn = xn+1y. We prove that every strongly π -regular ideal of a ring is a B-ideal. An ideal I is periodic provided that for any x ∈ I there exist two distinct m, n ∈ N such that xm = xn. Furthermore, we prove that an ideal I of a ring R is periodic if and only if I is strongly π -regular and for any u ∈ U(I), u-1 ∈ ℤ[u]. © 2014 Korean Mathematical Society.