Periodicity and J-Clean-like Rings

Abstract

A ring R is periodic provided that for any a ∈ R there exist distinct elements m, n ∈ N such that a m = a n . We shall prove that periodicity is inherited by all generalized matrix rings. A ring R is called strongly periodic if for any a ∈ R there exists a potent p ∈ R such that a − p is in its prime radical and ap = pa. A ring R is J-clean-like if for any a ∈ R there exists a potent p ∈ R such that a − p is in its Jacobson radical. Furthermore, we completely determine the connections between strongly periodic rings and periodic rings. The relations among J-clean-like rings and these rings are also obtained.