On power maps over periodic rings
Charles Burnette – GJM, Volume 10, Issue 1 (2025), 5-16.
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- juillet 29, 2025
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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, elementary 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 any 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, \Z/n\Z, endomorphism rings of finite abelian groups, and matrix rings over finite fields.
Milestones:
Received: December 12, 2024
Revised: June 08, 2025
Accepted: June 10, 2025
Published: July 25, 2025
Authors:
Charles BurnetteDepartment of Mathematics, Xavier University of Louisiana,
New Orleans, Louisiana 70125-1098, USA