Let $d_1,\ldots,d_r$ be pairwise relatively prime positive square-free integers, with $d_j\geq2$ for all $1\leq j\leq r$. Using elementary matrices and combinatorial mathematics, we give a complete classification of the irreducible $\mathbb{Z}_+$-modules over the domain $\mathbb{Z}[d^{\frac{1}{N_1}}_1,d^{\frac{1}{N_2}}_2,\dots,d^{\frac{1}{N_r}}_r]$, where $N_i \geq2$ for all $ 1\leq i \leq r$. Furthermore, we explicitly construct all of these irreducible $\mathbb{Z}_+$-modules.
Let $d_1,d_2,\ldots, d_r\ge 2$ be $m$-th power free positive integers. In this paper, we provide explicit representations of all irreducible $\mathbb{Z}_+$-modules over $\mathbb{Z}[\sqrt[m]{d_1},\sqrt[m]{d_2},\ldots,\sqrt[m]{d_r}]$, where $\mathbb{Z}_+$ denotes the semiring of non-negative integers.
Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerica...
Let $\mathbb{Z}[1/n] := \{a/n^r: a \in \mathbb{Z},r \in \mathbb{Z} \geq 0\}$ be the euclidean domain obtained from the ring of integers $\mathbb{Z}$ by localizing at $n$. Moreover, let $\text{SL}_2(\mathbb{Z}[1/n])$ be the group of all invertible 2-by-2 matrices of determinant one with entries in $\mathbb{Z}[1/n]$. The...
For a prime number $p>2$, let $\bm{0} \in D \subset \mathbb{Z}^n$ be a $p$-element digit set satisfying $ \mathcal{Z}(\widehat{\delta}_D) =\cup_{j=1}^{p-1}(\frac{j}{p}\bm{a}+\mathbb{Z}^{n}) $ for some \( \bm{a} \in \{ (i_1, \dots, i_n)^t : i_k \in [1, p-1] \cap \mathbb{Z}, 1\leq k\leq n \} \), where $\mathcal{Z}(\wideh...
The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. In this paper, first, we obtain the characteristic polynomial of $\mathscr{P}(\mathbb{Z}_m \time...
Komal Kumari, P. Panigrahi· Journal of Algebra Combinato...· 0 citations
Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, and let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$. Let $\mathrm{H}_{1,n-1}=\{\mathrm{diag}(g_1,g_2)\in\mathrm{G}_1,\ g_2\in\mathrm{G}_{n-1}\}$ and, for $s \in \mathbb{R}$, define $\chi_s(\mathrm{diag...
Prem Dagar, Hariom Sharma· 0 citations
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