Exponent of local ring extensions of Galois rings and digraphs of the $k$th power mapping

Authors : İttiwat TOCHAROENIRATTISAI, Yotsanan MEEMARK

Pages : 223-234

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, we consider a local extension $R$ of the Galois ring of the form $GRinsert ignore into journalissuearticles values(p^{n},d);[x]/insert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(x);^{a});$, where $n,d$, and $a$ are positive integers; $p$ is a prime; and $finsert ignore into journalissuearticles values(x);$ is a monic polynomial in $GRinsert ignore into journalissuearticles values(p^{n},d);[x]$ of degree $r$ such that the reduction $\overline{f}insert ignore into journalissuearticles values(x);$ 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 $G^{insert ignore into journalissuearticles values(k);}insert ignore into journalissuearticles values(R);$ 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 $G^{insert ignore into journalissuearticles values(k);}_{2}insert ignore into journalissuearticles values(R);$ such as indeg${}^{k} 0$ and maximum distance for $G^{insert ignore into journalissuearticles values(k);}_{2}insert ignore into journalissuearticles values(R);$ are determined by the nilpotency of maximal ideal $M$ of $R$.
Keywords : Finite chain rings, Galois rings, symmetric digraphs