Turkish Journal of Mathematics
DOI
10.3906/mat-1601-134
Abstract
In this paper, we consider a local extension $R$ of the Galois ring of the form $GR(p^{n},d)[x]/(f(x)^{a})$, where $n,d$, and $a$ are positive integers; $p$ is a prime; and $f(x)$ is a monic polynomial in $GR(p^{n},d)[x]$ of degree $r$ such that the reduction $\overline{f}(x)$ in $\mathbb{F}_{p^{d}}[x]$ is irreducible. We establish the exponent of $R$ without complete determination of its unit group structure. We obtain better analysis of the iteration graphs $G^{(k)}(R)$ induced from the $k$th power mapping including the conditions on symmetric digraphs. In addition, we work on the digraph over a finite chain ring $R$. The structure of $G^{(k)}_{2}(R)$ such as indeg${}^{k} 0$ and maximum distance for $G^{(k)}_{2}(R)$ are determined by the nilpotency of maximal ideal $M$ of $R$.
Keywords
Finite chain rings, Galois rings, symmetric digraphs
First Page
223
Last Page
234
Recommended Citation
TOCHAROENIRATTISAI, ITTIWAT and MEEMARK, YOTSANAN
(2017)
"Exponent of local ring extensions of Galois rings and digraphs of the $k$th power mapping,"
Turkish Journal of Mathematics: Vol. 41:
No.
2, Article 1.
https://doi.org/10.3906/mat-1601-134
Available at:
https://journals.tubitak.gov.tr/math/vol41/iss2/1
