Let R be a ring, and let N and C denote the set of nilpotents and the center of R , respectively. R is called generalized periodic if for every x ∈ R \ ( N ⋃ C ) , there exist distinct positive integers m , n of opposite parity such that x n − x m ∈ N ⋂ C . We prove that a generalized periodic ring always has the set N of nilpotents forming an ideal in R . We also consider some conditions which imply the commutativity of a generalized periodic ring.