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- W4288804748 abstract "A ring $R$ is called weakly periodic if every $x in R$ can be written in the form $x = a + b,$ where $a$ is nilpotent and $b^m = b$ for some integer $m > 1.$ The aim of this note is to consider when a nonzero nilpotent element $r$ is the period of some power map $f(x) = x^n,$ in the sense that $f(x + r) = f(x)$ for all $x in R,$ and how this relates to the structure of weakly periodic rings. In particular, we provide a new proof of the fact that weakly periodic rings with central and torsion nilpotent elements are periodic commutative torsion rings. We also prove that $x^n$ is periodic over such rings whenever $n$ is not coprime with each of the additive orders of the nilpotent elements. These are in fact the only periodic power maps over finite commutative rings with unity. Finally, we describe and enumerate the distinct power maps over Corbas $(p, k, phi)$-rings, Galois rings, $mathbb{Z}/nmathbb{Z},$ and matrix rings over finite fields." @default.
- W4288804748 created "2022-07-30" @default.
- W4288804748 creator A5034701168 @default.
- W4288804748 date "2022-07-28" @default.
- W4288804748 modified "2023-09-24" @default.
- W4288804748 title "On power maps over weakly periodic rings" @default.
- W4288804748 doi "https://doi.org/10.48550/arxiv.2207.14283" @default.
- W4288804748 hasPublicationYear "2022" @default.
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