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Mathematics
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New Advances in Algebra, Ring Theory and Homological Algebra
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New Advances in Algebra, Ring Theory and Homological Algebra

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

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 9568

Special Issue Editors

Prof. Dr. Juan Ramón García Rozas

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Guest Editor
Dept. de Matemáticas, Universidad de Almería, La Cañada de San Urbano S/N, 04120 Almería, Spain
Interests: module theory; homological algebra; quivers representation theory
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Luis Oyonarte Alcalá

E-Mail Website
Guest Editor
Dept. de Matemáticas, Universidad de Almería, La Cañada de San Urbano S/N, 04120 Almería, Spain
Interests: module theory; homological algebra; quivers representation theory
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Driss Bennis

E-Mail Website
Guest Editor
Dépt. de Mathématiques, Université Mohammed 5 de Rabat, 4 Avenue Ibn Batouta BP 1014 RP, Rabat, Morocco
Interests: module theory; homological algebra; quivers representation theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

The Development of Associative Algebra during the last century has resulted in the emergence of numerous theories or specialties that have given solution to many of the needs of the society we live in, increasingly developed from the technological point of view.

These needs fall into two broad groups: purely technological needs, and theoretical needs associated with developments in both applied algebra and other branches of mathematics. After all, it is not unreasonable to think that algebra is something like the "mathematics of mathematics".

There are many branches of algebra whose contributions solve problems posed by the scientific challenges arising from the advancement of technology. Two of them also stand out for their popularity in society: Cryptography and Coding Theory.

And from the theoretical point of view it is remarkable the momentum that some disciplines have had in the last 20 years. Thus, Homological Algebra has been given a big push with the emergence of the different classes of Gorenstein modules, and especially in recent years, of the relative Gorenstein modules. And the emergence of Hopf Algebras has made a huge impact on many branches of mathematics and physics. And of course, one cannot forget very active branches with immense applications at all times: Module Theory and Quivers Representation Theory.

Thus, we present this special issue of Mathematics as a tool to show recent and interesting results on the branches of Homological Algebra, Module Theory, Quivers Representation Theory, Hopf Algebras, Cryptography and Coding Theory.

Prof. Juan Ramón García Rozas
Prof. Luis Oyonarte Alcalá
Prof. Driss Bennis
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at mdpi.longhoe.net by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Covers, envelopes and cotorsion pairs
  • Relative Gorenstein modules and objects
  • Gorenstein dimensions
  • Grothendieck categories
  • Rings and modules
  • (Sub)projectivity, (sub)injectivity, (sub)flatness domains and extensions
  • Complexes of modules
  • Hopf Algebras
  • Algebraic coding theory
  • Cryptography

Published Papers (6 papers)

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Research

23 pages, 317 KiB  
Article
Multiplier Hopf Coquasigroup: Motivation and Biduality
by Tao Yang
Mathematics 2022, 10(21), 4006; https://doi.org/10.3390/math10214006 - 28 Oct 2022
Cited by 1 | Viewed by 815
Abstract
Inspired by the multiplier Hopf algebra theory introduced by A. Van Daele, this paper introduces a new algebraic structure, a multiplier Hopf coquasigroup, by constructing the integral dual of an infinite-dimensional Hopf quasigroup with faithful integrals. Then, it shows that the biduality theorem [...] Read more.
Inspired by the multiplier Hopf algebra theory introduced by A. Van Daele, this paper introduces a new algebraic structure, a multiplier Hopf coquasigroup, by constructing the integral dual of an infinite-dimensional Hopf quasigroup with faithful integrals. Then, it shows that the biduality theorem also holds for Hopf quasigroups and multiplier Hopf coquasigroups of the discrete type. Full article
(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)
7 pages, 261 KiB  
Article
On QTAG-Modules Having All N-High Submodules h-Pure
by Ayazul Hasan and Jules Clement Mba
Mathematics 2022, 10(19), 3523; https://doi.org/10.3390/math10193523 - 27 Sep 2022
Cited by 2 | Viewed by 1033
Abstract
The paper is concerned with h-pure-N-high submodules of QTAG-modules. Here, we characterize the submodules N of an h-reduced QTAG-module for which all h-pure-N-high submodules are bounded. We also [...] Read more.
The paper is concerned with h-pure-N-high submodules of QTAG-modules. Here, we characterize the submodules N of an h-reduced QTAG-module for which all h-pure-N-high submodules are bounded. We also discuss some interesting properties of subsocles and consequently give a characterization of the direct sum of uniserial modules. Full article
(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)
18 pages, 471 KiB  
Article
Public Key Protocols over Skew Dihedral Group Rings
by Javier de la Cruz, Edgar Martínez-Moro and Ricardo Villanueva-Polanco
Mathematics 2022, 10(18), 3343; https://doi.org/10.3390/math10183343 - 15 Sep 2022
Viewed by 1492
Abstract
This paper introduces skew dihedral group rings and their applications for public-key cryptography. We present a specific skew group ring that is the underlying algebraic platform for our cryptographic constructions. We then build a two-party key exchange protocol and present an analysis of [...] Read more.
This paper introduces skew dihedral group rings and their applications for public-key cryptography. We present a specific skew group ring that is the underlying algebraic platform for our cryptographic constructions. We then build a two-party key exchange protocol and present an analysis of its security. We then exploit it to derive a group key agreement protocol, a probabilistic public-key scheme, and a key encapsulation mechanism. In addition to the security analysis of our cryptographic constructions, we present a proof-of-concept implementation. Full article
(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)
16 pages, 318 KiB  
Article
Secure Group Communications Using Twisted Group Rings
by María Dolores Gómez Olvera, Juan Antonio López Ramos and Blas Torrecillas Jover
Mathematics 2022, 10(16), 2845; https://doi.org/10.3390/math10162845 - 10 Aug 2022
Cited by 1 | Viewed by 1143
Abstract
In this paper we introduce a Group Key Management protocol following the idea of the classical protocol that extends the well-known Diffie–Hellman key agreement to a group of users. The protocol is defined in a non-commutative setting, more precisely, in a twisted dihedral [...] Read more.
In this paper we introduce a Group Key Management protocol following the idea of the classical protocol that extends the well-known Diffie–Hellman key agreement to a group of users. The protocol is defined in a non-commutative setting, more precisely, in a twisted dihedral group ring. The protocol is defined for an arbitrary cocycle, extending previous key agreements considered for two users. The main objective of this work is to show that there is no lack of security derived from the fact that a larger amount of public information is known by an external observer. Full article
(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)
28 pages, 618 KiB  
Article
Relative Gorenstein Dimensions over Triangular Matrix Rings
by Driss Bennis, Rachid El Maaouy, Juan Ramón García Rozas and Luis Oyonarte
Mathematics 2021, 9(21), 2676; https://doi.org/10.3390/math9212676 - 22 Oct 2021
Cited by 2 | Viewed by 1647
Abstract
Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first [...] Read more.
Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension. Full article
(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)
34 pages, 483 KiB  
Article
Long Dimodules and Quasitriangular Weak Hopf Monoids
by José Nicanor Alonso Álvarez, José Manuel Fernández Vilaboa and Ramón González Rodríguez
Mathematics 2021, 9(4), 424; https://doi.org/10.3390/math9040424 - 21 Feb 2021
Cited by 1 | Viewed by 1583
Abstract
In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules HBLong is monoidal. Moreover, if H is [...] Read more.
In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules HBLong is monoidal. Moreover, if H is quasitriangular and B coquasitriangular, we also prove that HBLong is braided. As a consequence of this result, we obtain that if H is triangular and B cotriangular, HBLong is an example of a symmetric monoidal category. Full article
(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)
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