Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.37150

Finite Matrix Rings of Even Size Are Determined by Their Commuting Graphs

Hamid Shahverdi·Vahid Shahverdi
View original source ↗Read source document ↗Export metadata ↓

Abstract

The commuting graph of a noncommutative ring has its noncentral elements as vertices, with two distinct vertices adjacent when they commute. Akbari, Ghandehari, Hadian and Mohammadian conjectured that a full matrix ring over a finite field is determined, among rings with identity, by its commuting graph. In this paper, we prove the conjecture for every even matrix size.

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hamid Shahverdi, Vahid Shahverdi. 2026-10-27. Finite Matrix Rings of Even Size Are Determined by Their Commuting Graphs. https://arxiv.org/abs/2610.37150

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Spectrum in alternative topological algebras and a new look at old theorems

In this paper, we consider real and complex algebras as well as algebras over general fields. In Section 2, we revisit and prove several results on (quadratic) algebras over general fields. As an example, we demonstrate that a quadratic algebra over a field of characteristic not $2$ is flexible if and only if it is proper--a concept introduced in this paper. In Section 3, we show how to develop the spectral theory in the context of complex (resp. real) one-sided alternative topological algebras. Along the way, spectral theory is used to prove several topological counterparts of the well-known theorems of Frobenius, Hurwitz, Gelfand-Mazur, and Zorn. This is done, for example, in the context of left (resp. right) alternative topological algebras whose duals separate their elements. In Section 4, we consider real and complex algebras in various topological settings and reconsider and prove several results. For instance, it is shown that given a $ 1 < k \in \mathbb{N}$, on any locally complex algebra, there exists a unique nonzero vector space norm, say, $\|.\|$, satisfying the identity $\|a^k\| = \|a\|^k$ on the algebra. In Section 5, among other things, we revisit and slightly strengthen the celebrated theorems of Frobenius, Zorn, Gelfand-Mazur, and Hurwitz, and also give slight extensions of their topological counterparts, e.g., theorems of Albert, Kaplansky, and Urbanik-Wright to name a few, in several

math.RA↗
arXiv2026

Positive semidefiniteness of $h$-Shuhan matrices (analogs of Borcherds Cartan matrices) and associated determinant sequences

We introduce an $h$-Shuhan matrix as a real square matrix $H=(h_{ij})$ with a common diagonal entry $h\ge0$, nonpositive integral off-diagonal entries, and the following Cartan-type condition on each unordered pair $\{i,j\}$: either $h_{ij}=h_{ji}$, or $\max\{h_{ij},h_{ji}\}=-1$ and $\min\{h_{ij},h_{ji}\}\le -2$. These conditions imply that the scalar shift $H+(2-h)I_n$ is a generalized Cartan matrix; not every generalized Cartan matrix arises from an $h$-Shuhan matrix in this way. We classify, for $0\le h<2$, all indecomposable $h$-Shuhan matrices that are positive semidefinite, generalized positive semidefinite, or virtual positive semidefinite (the last notion means that all principal minors are nonnegative). For $h<2$, virtual positive semidefiniteness forces $H+(2-h)I_n$ to be of finite Cartan type, the positive-type case of Vinberg's trichotomy (1971). At $h=2$, we describe the boundary case in terms of finite and affine Cartan matrices. The associated determinant recurrences hold for general $h$. We study congruences for integral $h$ and asymptotic and Fibonacci-type identities for $h>2$.

math.RA↗
arXiv2026

The Dixmier problem for skew PBW extensions and rings

In this paper we discuss for skew $PBW$ extensions the famous Dixmier problem formulated by Jacques Dixmier in 1968. The skew $PBW$ extensions are noncommutative rings of polynomial type and covers several algebras and rings arising in mathematical physics and noncommutative algebraic geometry. For this purpose, we introduce the Dixmier algebras and we will study the Dixmier problem for algebras over commutative rings, in particular, for $\mathbb{Z}$-algebras, i.e., for arbitrary rings. The results are focused on the investigation of the Dixmier problem for matrix algebras, product of algebras, tensor product of algebras and also on the Dixmier question for the following particular key skew $PBW$ extension: Let $K$ be a field of characteristic zero and let $\mathcal{CSD}_n(K)$ be the $K$-algebra generated by $n\geq 2$ elements $x_1,\dots,x_n$ subject to relations $$x_jx_i=x_ix_j+d_{ij}, \ for \ all \ 1\leq i<j\leq n, \ with \ d_{ij}\in K-\{0\}$$. We prove that the algebra $\mathcal{CSD}_n(K)$ is central and simple. In the last section we present a matrix-computational approach to the problem formulated by Jacques Dixmier and also we compute some concrete nontrivial examples of automorphisms of the first Weyl algebra $A_1(K)$ and $\mathcal{CSD}_n(K)$ using the MAPLE library SPBWE developed for the first author. We compute the inverses of these automorphisms, and for $A_1(K)$, its factorization through some elementary automorphisms. For $n$ odd, we found some endomorphisms of $\mathcal{CSD}_n(K)$ that are not automorphisms. We conjecture that $\mathcal{CSD}_n(K)$ is Dixmier when $n$ is even.

math.RA↗