  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.
