Special Issue "Latest Trends in Noncommutative Algebra"

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: closed (20 May 2022) | Viewed by 2706

Special Issue Editor

1. Department of Mathematics, Swansea University, Bay Campus, Fabian Way, Swansea SA1 8EN, UK
2. Department of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland
Interests: noncommutative geometry; corings and comodules; Hopf algebras; category theory; algebraic structures
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Special Issue Information

Dear Colleagues,

The origin of noncommutative algebra goes back to the mid 19th century when Hamilton discovered quaternions. Later, Wedderburn, Noether and Artin laid down the foundation of the theory of noncommutative rings. In the past few decades, many new topics have emerged on the interface of noncommutative algebra with algebraic geometry, operator algebra and physics, such as Hopf algebra, noncommutative Calabi–Yau algebra, quantum cluster algebra, and Leavitt path algebra. In the area of the study of modules over noncommutative rings, there have been a lot of advances in the past few decades, i.e., the theory of approximations of modules, generalizations of homological properties, and the theory of purity.

In this special volume, I would like to invite contributions that highlight recent developments in the area of noncommutative rings and modules over them. These may include tools and techniques that are categorical, combinatorial or homological. Contributions leaning toward applications in coding theory are welcome too. 

Prof. Dr. Tomasz Brzezinski
Guest Editor

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  • noncommutative rings
  • module theory
  • noncommutative algebraic geometry
  • Hopf algebra, quantum cluster algebra, Leavitt path algebra
  • coding theory

Published Papers (1 paper)

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On r-Noncommuting Graph of Finite Rings
Axioms 2021, 10(3), 233; https://doi.org/10.3390/axioms10030233 - 19 Sep 2021
Cited by 3 | Viewed by 1420
Let R be a finite ring and rR. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if [...] Read more.
Let R be a finite ring and rR. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]r and [x,y]r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n6. Full article
(This article belongs to the Special Issue Latest Trends in Noncommutative Algebra)
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