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Let $FG$ be a group algebra of a group $G$ over a field $F$ and $𝒰\left(FG\right)$ the unit group of $FG$. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group $G$ with order $21$ over any finite field of characteristic $3$ is established. We also characterize the structure of the unit group of $F{A}_4$ over any finite field of characteristic $3$ and the structure of...

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...

For a finite commutative ring $R$ and a positive integer $k⩾2$, 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...

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 ${\mathbb{Z}}_n\left[\mathrm{i}\right]$.

An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are...