Periodicity and J-Clean-like Rings

TL;DR

This paper investigates the properties of periodic and J-clean-like rings, establishing inheritance results, defining strongly periodic rings, and exploring their interrelations within generalized matrix rings.

Contribution

It introduces the concept of strongly periodic rings and characterizes J-clean-like rings, expanding the understanding of ring structures and their interconnections.

Findings

Periodic rings are inherited by certain generalized matrix rings.

Strongly periodic rings are fully characterized and related to periodic rings.

Relations among J-clean-like rings and other ring classes are established.

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.