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Some new characterizations of periodic rings. (English) Zbl 1460.16042

A ring \(R\) is called \(\pi \)-UU if, for every unit \(u\), there exists some positive integer \(n\) such that \(u^{n}=1+b\), where \(b\) is nilpotent in \(R \). A ring \(R\) is potent if, for every \(a\in R\), \(a^{n}=a\) for some integer \(n\geq 2\), and \(R\) is periodic if, for every \(a\in R\), there exist distinct positive integers \(m\) and \(n\) such that \(a^{m}=a^{n}\).
Potent rings are periodic and periodic rings are \(\pi \)-UU.
From the introduction:“ In this article, some examples and basic properties of \(\pi \)-UU rings are investigated. Properties such as being \(\pi \)-UU, strongly \( \pi \)-regularity and strongly nil cleanness are subsequently applied to characterize periodic rings as some equivalent statements are obtained. Further, by extending periodic rings to their *-versions, the notion of *-periodic rings is introduced. Various characterizations of *-periodic rings are provided. In particular, it is shown that a ring \(R\) is *-periodic iff \(R\) is periodic and idempotents of \(R\) are projections iff \(R\) is a strongly \(\pi \)-regular \( \pi \)-UU ring and idempotents of \(R\) are projections iff \(R\) is an abelian \( \pi \)-UU ring, \(R/J(R)\) is *-regular and \(J(R)\) is nil.”

MSC:

16U99 Conditions on elements
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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