### A class of finite rings

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Let $R$ be a commutative ring with nonzero identity and $J\left(R\right)$ the Jacobson radical of $R$. The Jacobson graph of $R$, denoted by ${\U0001d50d}_{R}$, is defined as the graph with vertex set $R\setminus J\left(R\right)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1-xy$ is not a unit of $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface ${S}_{n}$. In this paper, we investigate the genus number of the compact Riemann surface in which ${\U0001d50d}_{R}$ can be embedded and explicitly...

Let $R$ be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.

Let ${\mathbb{Z}}_{n}\left[\mathrm{i}\right]$ be the ring of Gaussian integers modulo $n$. We construct for ${\mathbb{Z}}_{n}\left[\mathrm{i}\right]$ a cubic mapping graph $\Gamma \left(n\right)$ whose vertex set is all the elements of ${\mathbb{Z}}_{n}\left[\mathrm{i}\right]$ and for which there is a directed edge from $a\in {\mathbb{Z}}_{n}\left[\mathrm{i}\right]$ to $b\in {\mathbb{Z}}_{n}\left[\mathrm{i}\right]$ if $b={a}^{3}$. This article investigates in detail the structure of $\Gamma \left(n\right)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in ${\Gamma}_{1}\left(n\right)$ is obtained and we also examine when a vertex lies on a $t$-cycle of ${\Gamma}_{2}\left(n\right)$, where ${\Gamma}_{1}\left(n\right)$ is induced by all the units of ${\mathbb{Z}}_{n}\left[\mathrm{i}\right]$ while ${\Gamma}_{2}\left(n\right)$ is induced by all the...

The article studies the cubic mapping graph $\Gamma \left(n\right)$ of ${\mathbb{Z}}_{n}\left[\mathrm{i}\right]$, the ring of Gaussian integers modulo $n$. For each positive integer $n>1$, the number of fixed points and the in-degree of the elements $\overline{1}$ and $\overline{0}$ in $\Gamma \left(n\right)$ are found. Moreover, complete characterizations in terms of $n$ are given in which ${\Gamma}_{2}\left(n\right)$ is semiregular, where ${\Gamma}_{2}\left(n\right)$ is induced by all the zero-divisors of ${\mathbb{Z}}_{n}\left[\mathrm{i}\right]$.

For a finite commutative ring $R$ and a positive integer $k\u2a7e2$, we construct an iteration digraph $G(R,k)$ whose vertex set is $R$ and for which there is a directed edge from $a\in R$ to $b\in R$ if $b={a}^{k}$. Let $R={R}_{1}\oplus ...\oplus {R}_{s}$, where $s>1$ and ${R}_{i}$ is a finite commutative local ring for $i\in \{1,...,s\}$. Let $N$ be a subset of $\{{R}_{1},\cdots ,{R}_{s}\}$ (it is possible that $N$ is the empty set $\varnothing $). We define the fundamental constituents ${G}_{N}^{*}(R,k)$ of $G(R,k)$ induced by the vertices which are of the form $\{({a}_{1},\cdots ,{a}_{s})\in R:{a}_{i}\in \mathrm{D}\left({R}_{i}\right)$ if ${R}_{i}\in N$, otherwise ${a}_{i}\in \mathrm{U}\left({R}_{i}\right),i=1,...,s\},$ where U$\left(R\right)$ denotes the unit group of $R$ and D$\left(R\right)$ denotes the zero-divisor set of $R$. We investigate...