Search Results
Abstract
Let R be an associative ring with unit and let N(R) denote the set of nilpotent elements of R. R is said to be stronglyπ-regular if for each x∈R, there exist a positive integer n and an element y∈R such that x n=x n +1 y and xy=yx. R is said to be periodic if for each x∈R there are integers m,n≥ 1 such that m≠n and x m=x n. Assume that the idempotents in R are central. It is shown in this paper that R is a strongly π-regular ring if and only if N(R) coincides with the Jacobson radical of R and R/N(R) is regular. Some similar conditions for periodic rings are also obtained.
1. Introduction and preliminaries Let 𝔄 be a complex Banach algebra with unit 1 and Jacobson radical ℜ . Recall that ℜ coincides with the intersection of the maximal modular left (right) ideals, with ℜ = 𝔄 if there are no such (and then
A ring R is called periodic if, for every x in R, there exist distinct positive integers m and n such that x m=x n. An element x is called potent if x k=x for some integer k≯1. A ring R is called weakly periodic if every x in R can bewritten in the form x=a+b for some nilpotent element a and some potent element b. A ring R is called weakly periodic-like if every x in R which is not in the center of R can be written in the form x=a+b, where a is nilpotent and b is potent. Our objective is to study the structure of weakly periodic-like rings, with particular emphasis on conditions which yield such rings commutative, or conditions which render the nilpotents N as an ideal of R and R/N as commutative.
