Rings close to periodic with applications to matrix, endomorphism and group rings

Abstract: We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is nil-clean for all $n\geq 1$ is paralleling to the corresponding implication for (Abelian, local) periodic rings. Besides, we study when the endomorphism ring $\mathrm{E}(G)$ of an Abelian group $G$ is periodic. Concretely, we establish that $\mathrm{E}(G)$ is periodic exactly when $G$ is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an Abelian group is strongly $m$-nil clean for some natural number $m$ thus refining an "old" result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if $R$ is a right (resp., left) perfect periodic ring and $G$ is a locally finite group, then the group ring $RG$ is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated.