Periodicity and J-Clean-like Rings

Abstract: A ring $R$ is periodic provided that for any $a\ in R$ there exist distinct elements $m,n \in {\Bbb N}$ such that $a^m=a^n$. We shall prove that periodicity is inherited by a type of generalized matrix rings.We define strongly periodic rings and determine completely the connections between these ones and periodic rings. Moreover, we characterize J-clean like rings and obtain the relations among these rings.