论文标题
在弱周期性环上的电源图上
On power maps over weakly periodic rings
论文作者
论文摘要
如果可以用$ x = a + b，$ a $写入$ x = a $的每一个$ x \ in r $的每一个$ r $，则称为弱周期性，其中$ a $是nilpotent的，对于某些整数$ m> 1的$ b^m = b $ 1. $的目的是考虑何时非零的nilpotent元素$ r $何时$ r $ n $ r $何时$ f（x）$ f（x） f（x）$ for r，$中的所有$ x \，以及与弱周期性环的结构之间的关系。 特别是，我们提供了一个新的证据，证明具有中央和扭转元素的弱周期性环是周期性的扭转环。我们还证明，只要$ n $与nilpotent元素的每个添加剂订单都不是企业时，$ x^n $是定期的。实际上，这些是与有限的统一性环上的唯一定期功率图。最后，我们描述并枚举Corbas $（P，K，ϕ）$ - 戒指，Galois环，$ \ Mathbb {Z}/N \ Mathbb {Z}，$和矩阵环，有限字段上的矩阵环。
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 $(p, k, ϕ)$-rings, Galois rings, $\mathbb{Z}/n\mathbb{Z},$ and matrix rings over finite fields.
