Advanced Research in Pure and Applied Algebra

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "A: Algebra and Logic".

Deadline for manuscript submissions: 31 July 2025 | Viewed by 7321

Special Issue Editor


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Guest Editor
School of Mathematical Science, Heilongjiang University, Harbin 150080, China
Interests: lie theory; cluster algebra; brace

Special Issue Information

Dear Colleagues,

This Special Issue entitled “Advanced Research in Pure and Applied Algebra” endeavors to publish research papers of the highest quality, with an appeal to specialists in the field of linear and abstract algebra and to the broader mathematical community. We hope that the distinctive aspects of this Special Issue will bring the reader close to the current research in these fields and open avenues for more direct and less ambivalent approaches for these topics.

Our goal is to invite authors to present their original articles, as well as review articles, that will stimulate continuing efforts to develop new results in these areas of interest. We hope that this Special Issue will have a great impact on other people in their efforts to broaden their knowledge and investigation and help researchers to summarize the most recent developments and ideas in these fields.

This Special Issue invites authors to present their original articles that not only provide new results or methods but also may have a great impact on other people in their efforts to broaden their knowledge and investigation.

Prof. Dr. Xiaomin Tang
Guest Editor

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Keywords

  • lie algebra and lie superalgebra
  • associative algebra
  • representation theory
  • cluster algebra
  • rota–baxter algebra
  • group theory
  • braces
  • quantum algebra
  • matrix algebra

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Published Papers (9 papers)

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Research

12 pages, 247 KiB  
Article
On a Quotient Ring That Satisfies Certain Identities via Generalized Reverse Derivations
by Nawaf L. Alsowait, Mohammed Al-Shomrani, Radwan M. Al-omary and Zakia Z. Al-Amery
Mathematics 2025, 13(5), 870; https://doi.org/10.3390/math13050870 - 5 Mar 2025
Viewed by 382
Abstract
In this article, for a prime ideal ρ of an arbitrary ring ℜ, we study the commutativity of the quotient ring /ρ, whenever ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation that satisfies certain identities [...] Read more.
In this article, for a prime ideal ρ of an arbitrary ring ℜ, we study the commutativity of the quotient ring /ρ, whenever ℜ admits a generalized reverse derivation ϑ associated with a reverse derivation that satisfies certain identities in ρ. Additionally, we show that, for some cases, the range of the generalized reverse derivation ϑ lies in the prime ideal ρ. Moreover, we explore several consequences and special cases. Throughout, we provide examples to demonstrate that various restrictions in the assumptions of our results are essential. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
13 pages, 235 KiB  
Article
Lie Ideals and Homoderivations in Semiprime Rings
by Ali Yahya Hummdi, Zeliha Bedir, Emine Koç Sögütcü, Öznur Gölbaşı and Nadeem ur Rehman
Mathematics 2025, 13(4), 548; https://doi.org/10.3390/math13040548 - 7 Feb 2025
Viewed by 497
Abstract
Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of S. An additive mapping on S is defined as a homoderivation if [...] Read more.
Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of S. An additive mapping on S is defined as a homoderivation if (ab)=(a)(b)+(a)b+a(a) for all a,bS. In the present paper, we shall prove that is a commuting map on U if any one of the following holds: (i)(a˜1a˜2)+a˜1a˜2Z, (ii)(a˜1a˜2)a˜1a˜2Z, (iii)a˜1a˜2=0, (iv)a˜1a˜2=a˜1,a˜2, (v)a˜1,a˜2=0, (vi)a˜1,a˜2= (a˜1a˜2), (vii)a˜1(a˜2)±a˜1a˜2Z, (viii)a˜1(a˜2)±a˜2a˜1=0, (ix)a˜1(a˜2)±a˜1a˜2=0, (x)[(a˜1),a˜2]±a˜1a˜2=0, (xi)[(a˜1),a˜2]±a˜2a˜1=0, for all a˜1,a˜2U, where is a homoderivation on S. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
12 pages, 264 KiB  
Article
The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups
by Yanliang Cheng
Mathematics 2025, 13(3), 399; https://doi.org/10.3390/math13030399 - 25 Jan 2025
Viewed by 473
Abstract
In this paper, the equivalent standard forms of tropical idempotent strongly definite matrices are introduced. In particular, the observation of the equivalent standard forms of tropical idempotent normal matrices is given. An equivalence relation ρ on the set of all tropical idempotent normal [...] Read more.
In this paper, the equivalent standard forms of tropical idempotent strongly definite matrices are introduced. In particular, the observation of the equivalent standard forms of tropical idempotent normal matrices is given. An equivalence relation ρ on the set of all tropical idempotent normal matrices, which is relevant to their centralizer groups, is introduced and studied. It is proved that every ρ-class contains at least one strongly regular tropical idempotent normal matrix. Furthermore, a structural description of the centralizer groups of partial strongly regular tropical idempotent normal matrices is given. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
14 pages, 274 KiB  
Article
An Introduction to i-Commutative Rings
by Muhammad Saad, Usama A. Aburawash, Ahmed M. A. El-Sayed and Nour Nabil
Mathematics 2025, 13(2), 253; https://doi.org/10.3390/math13020253 - 14 Jan 2025
Viewed by 706
Abstract
In this paper, we introduce a new class of rings, called i-commutative rings, by extending the concept of commutative-like rings using idempotent elements. In particular, we study rings with the property that, whenever ab+cd is a nontrivial idempotent, then [...] Read more.
In this paper, we introduce a new class of rings, called i-commutative rings, by extending the concept of commutative-like rings using idempotent elements. In particular, we study rings with the property that, whenever ab+cd is a nontrivial idempotent, then ba+dc is idempotent. We explore the basic properties of these rings and their relations with other rings. Moreover, we provide some examples using matrices and describe the structure of the idempotent elements in these rings. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
14 pages, 286 KiB  
Article
The Category G-GrR-Mod and Group Factorization
by Rahmah Al-Omari and Mohammed Al-Shomrani
Mathematics 2024, 12(21), 3344; https://doi.org/10.3390/math12213344 - 25 Oct 2024
Viewed by 606
Abstract
In this work, we use the concept of G-weak graded rings and G-weak graded modules, which are based on grading by a set G of left coset representatives for the left action of a subgroup H of a finite X on [...] Read more.
In this work, we use the concept of G-weak graded rings and G-weak graded modules, which are based on grading by a set G of left coset representatives for the left action of a subgroup H of a finite X on X, to define the conjugation action of the set G and to generalize and prove some results from the literature. In particular, we prove that a G-weak graded ring R is strongly graded if and only if each G-weak graded R-module V is induced by an ReG-module. Moreover, we prove that the additive induction functor ()R and the restriction functor ()eG form an equivalence between the categories G-GrR-Mod and ReG-Mod when R is strongly G-weak graded. Furthermore, some related results and illustrative examples of G-weak graded R-modules and their morphisms are provided. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
14 pages, 448 KiB  
Article
The Connection between Der(Uq+(g)) and Der(Ur,s+(g))
by Yongyue Zhong and Xiaomin Tang
Mathematics 2024, 12(16), 2517; https://doi.org/10.3390/math12162517 - 15 Aug 2024
Viewed by 632
Abstract
To facilitate the parallel development of the structures and properties of two-parameter quantum groups with those of one-parameter quantum groups, this paper primarily elucidates the interrelations and distinctions between the derivation algebras of these two types of quantum groups. Additionally, we also proposed [...] Read more.
To facilitate the parallel development of the structures and properties of two-parameter quantum groups with those of one-parameter quantum groups, this paper primarily elucidates the interrelations and distinctions between the derivation algebras of these two types of quantum groups. Additionally, we also proposed a method for deriving the derivation algebra of one-parameter quantum groups from known two-parameter quantum group derivations, and vice versa. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
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10 pages, 260 KiB  
Article
Exploring Commutativity via Generalized (α, β)-Derivations Involving Prime Ideals
by Nawaf Alsowait, Radwan M. Al-omary, Zakia Al-Amery and Mohammed Al-Shomrani
Mathematics 2024, 12(15), 2325; https://doi.org/10.3390/math12152325 - 25 Jul 2024
Cited by 3 | Viewed by 849
Abstract
The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring /, where is a prime ideal in a ring . In particular, we are going to explore more general scenarios [...] Read more.
The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring /, where is a prime ideal in a ring . In particular, we are going to explore more general scenarios whenever a ring admits a generalized (α,β)-derivation associated with an (α,β)-derivation that satisfies certain criteria involving , where α and β are automorphisms on . Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
17 pages, 310 KiB  
Article
Cohomology and Crossed Modules of Modified Rota–Baxter Pre-Lie Algebras
by Fuyang Zhu and Wen Teng
Mathematics 2024, 12(14), 2260; https://doi.org/10.3390/math12142260 - 19 Jul 2024
Viewed by 1046
Abstract
The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with [...] Read more.
The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras. Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
12 pages, 274 KiB  
Article
The Ribbon Elements of the Quantum Double of Generalized Taft–Hopf Algebra
by Hua Sun, Yuyan Zhang, Ziliang Jiang, Mingyu Huang and Jiawei Hu
Mathematics 2024, 12(12), 1802; https://doi.org/10.3390/math12121802 - 10 Jun 2024
Viewed by 897
Abstract
Let s, t be two positive integers and k be an algebraically closed field with char (k)st. We show that the Drinfeld double D(st,t*cop) of [...] Read more.
Let s, t be two positive integers and k be an algebraically closed field with char (k)st. We show that the Drinfeld double D(st,t*cop) of generalized Taft–Hopf algebra st,t*cop has ribbon elements if and only if t is odd. Moreover, if s is even and t is odd, then D(st,t*cop) has two ribbon elements, and if both s and t are odd, then D(st,t*cop) has only one ribbon element. Finally, we compute explicitly all ribbon elements of D(st,t*cop). Full article
(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)
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