New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition

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

Deadline for manuscript submissions: closed (30 November 2024) | Viewed by 13076

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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

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

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 offered solutions to many of the needs of the society we live in, increasingly developed from a 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.

Additionally, from a theoretical point of view, the momentum that some disciplines have had in the last 20 years is remarkable. 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. Moreover, the emergence of Hopf Algebras has had a huge impact on many branches of mathematics and physics—and, of course, one cannot forget the very active branches of module theory and quivers representation theory, with immense applications at all times.

Thus, we are pleased to present this Special Issue of Mathematics as a tool to share recent and interesting results in the branches of homological algebra, module theory, quivers representation theory, Hopf algebras, cryptography, and coding theory.

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

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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
  • representations of quivers by modules.

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Published Papers (12 papers)

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19 pages, 351 KiB  
Article
On Lusztig’s Character Formula for Chevalley Groups of Type Al
by Sherali S. Ibraev, Larissa Kainbaeva, Gulzat M. Yensebayeva, Anar A. Ibrayeva, Manat Z. Parmenova and Gulnur K. Yeshmurat
Mathematics 2024, 12(23), 3791; https://doi.org/10.3390/math12233791 - 30 Nov 2024
Viewed by 359
Abstract
For a Chevalley group G over an algebraically closed field K of characteristic p>0 with the irreducible root system R, Lusztig’s character formula expresses the formal character of a simple G-module by the formal characters of the Weyl modules [...] Read more.
For a Chevalley group G over an algebraically closed field K of characteristic p>0 with the irreducible root system R, Lusztig’s character formula expresses the formal character of a simple G-module by the formal characters of the Weyl modules and the values of the Kazhdan–Lusztig polynomials at 1. It is known that, for a sufficiently large characteristic p of the field K, Lusztig’s character formula holds. The known lower bound of the characteristic p is much larger than the Coxeter number h of the root system R. Observations show that for simple modules with restricted highest weights of small Chevalley groups such as those of types A1,A2, A3,B2, B3, and C3, Lusztig’s character formula holds for all ph. For large Chevalley groups, no other examples are known. In this paper, for G of type Al, we give some series of simple modules for which Lusztig’s character formula holds for all ph. Using this result, we compute the cohomology of G with coefficients in these simple modules. To prove the results, Jantzen’s filtration properties for Weyl modules and the properties of Kazhdan–Lusztig polynomials are used. Full article
12 pages, 229 KiB  
Article
Characterizations of Commutativity of Prime Ring with Involution by Generalized Derivations
by Mingxing Sui and Quanyuan Chen
Mathematics 2024, 12(14), 2286; https://doi.org/10.3390/math12142286 - 22 Jul 2024
Viewed by 627
Abstract
In the paper, we investigate the commutativity of a two-torsion free prime ring R provided with generalized derivations, and some well-known results that characterize the commutativity of prime rings through generalized derivations have been generalized. Moreover, we provide some examples to testify that [...] Read more.
In the paper, we investigate the commutativity of a two-torsion free prime ring R provided with generalized derivations, and some well-known results that characterize the commutativity of prime rings through generalized derivations have been generalized. Moreover, we provide some examples to testify that the assumed restriction in our theorems cannot be omitted. Full article
11 pages, 265 KiB  
Article
Linear Generalized n-Derivations on C-Algebras
by Shakir Ali, Amal S. Alali and Vaishali Varshney
Mathematics 2024, 12(10), 1558; https://doi.org/10.3390/math12101558 - 16 May 2024
Viewed by 1413
Abstract
Let n2 be a fixed integer and A be a C-algebra. A permuting n-linear map G:AnA is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation [...] Read more.
Let n2 be a fixed integer and A be a C-algebra. A permuting n-linear map G:AnA is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D:AnA such that Gς1,ς2,,ςiςi,,ςn=Gς1,ς2,,ςi,,ςnςi+ςiD(ς1,ς2,,ςi,,ςn) holds ∀ςi,ςiA. In this paper, we investigate the structure of C-algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity. Full article
30 pages, 383 KiB  
Article
(X,Y)-Gorenstein Categories, Associated (Global) Homological Dimensions and Applications to Relative Foxby Classes
by Enrique Duarte, Juan Ramón García Rozas, Hanane Ouberka and Luis Oyonarte
Mathematics 2024, 12(8), 1130; https://doi.org/10.3390/math12081130 - 9 Apr 2024
Viewed by 1007
Abstract
Recently, Gorenstein dimensions relative to a semidualizing module have been the subject of numerous studies with interesting extensions of the classical homological dimensions. Although all these studies share the same direction, a common basis, and similar final goals, there is no common framework [...] Read more.
Recently, Gorenstein dimensions relative to a semidualizing module have been the subject of numerous studies with interesting extensions of the classical homological dimensions. Although all these studies share the same direction, a common basis, and similar final goals, there is no common framework encompassing them as parts of a whole, progressing, on different fronts, towards the same end. We provide this general and global framework in the context of abelian categories, standardizing terminology and notation: we establish a general context by defining Gorenstein categories relative to two classes of objects ((X,Y)-Gorenstein categories, denoted G(X,Y)), and carry out a study of the homological dimensions associated with them. We prove, under some mild standard conditions, the corresponding version of the Comparison Lemma that ensures the consistency of a homological-dimension theory. We show that Ext functors can be used as tools to compute these G(X,Y)-dimensions, and we compare the dimensions obtained using the classes G(X) with those computed using G(X,Y). We also initiate a research of the global dimensions obtained with these classes G(X,Y) and find conditions for them to be finite. Finally, we show that these classes of Gorenstein objects are closely and interestingly related to the Foxby classes induced by a pair of functors. Namely, we prove that the Auslander and Bass classes are indeed G(X,Y) categories for some specific classes X and Y. Full article
20 pages, 847 KiB  
Article
Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions
by Ab Hamid Kawa, Turki Alsuraiheed, S. N. Hasan, Shakir Ali and Bilal Ahmad Wani
Mathematics 2023, 11(23), 4770; https://doi.org/10.3390/math11234770 - 25 Nov 2023
Cited by 1 | Viewed by 916
Abstract
Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:AA is a Lie-type higher derivation. In continuation of the rigorous and [...] Read more.
Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:AA is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕm:AA and an additive higher map ζm:AZ(A), which annihilates every (n1)th commutator pn(S1,S2,,Sn) with S1S2=0 such that Lm(S)=ϕm(S)+ζm(S)forallSA. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results. Full article
12 pages, 279 KiB  
Article
Nonlinear Skew Lie-Type Derivations on ∗-Algebra
by Md Arshad Madni, Amal S. Alali and Muzibur Rahman Mozumder
Mathematics 2023, 11(18), 3819; https://doi.org/10.3390/math11183819 - 6 Sep 2023
Viewed by 1243
Abstract
Let A be a unital ∗-algebra over the complex fields C. For any H1,H2A, a product [H1,H2]=H1H2H2H1* [...] Read more.
Let A be a unital ∗-algebra over the complex fields C. For any H1,H2A, a product [H1,H2]=H1H2H2H1* is called the skew Lie product. In this article, it is shown that if a map ξ : AA (not necessarily linear) satisfies ξ(Pn(H1,H2,,Hn))=i=1nPn(H1,,Hi1,ξ(Hi),Hi+1,,Hn)(n3) for all H1,H2,,HnA, then ξ is additive. Moreover, if ξ(ie2) is self-adjoint, then ξ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type I1. Full article
22 pages, 353 KiB  
Article
Acyclic Complexes and Graded Algebras
by Chaoyuan Zhou
Mathematics 2023, 11(14), 3167; https://doi.org/10.3390/math11143167 - 19 Jul 2023
Viewed by 1035
Abstract
We already know that the noncommutative N-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and [...] Read more.
We already know that the noncommutative N-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative N-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra. Full article
20 pages, 363 KiB  
Article
Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications
by Shakir Ali, Turki M. Alsuraiheed, Mohammad Salahuddin Khan, Cihat Abdioglu, Mohammed Ayedh and Naira N. Rafiquee
Mathematics 2023, 11(14), 3117; https://doi.org/10.3390/math11143117 - 14 Jul 2023
Cited by 1 | Viewed by 1088
Abstract
A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to [...] Read more.
A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman’s theorem. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime A with involution admits a nonzero ∗-centralizing derivation, then A contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research. Full article
12 pages, 302 KiB  
Article
Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties
by Daniel Camazón Portela and Juan Antonio López Ramos
Mathematics 2023, 11(14), 3079; https://doi.org/10.3390/math11143079 - 12 Jul 2023
Viewed by 1073 | Correction
Abstract
The aim of this article is to give lower bounds on the parameters of algebraic geometric error-correcting codes constructed from projective bundles over Deligne–Lusztig surfaces. The methods based on an intensive use of the intersection theory allow us to extend the codes previously [...] Read more.
The aim of this article is to give lower bounds on the parameters of algebraic geometric error-correcting codes constructed from projective bundles over Deligne–Lusztig surfaces. The methods based on an intensive use of the intersection theory allow us to extend the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. General bounds are obtained for the case of projective bundles of rank 2 over standard Deligne–Lusztig surfaces, and some explicit examples coming from surfaces of type A2 and 2A4 are given. Full article
8 pages, 272 KiB  
Article
On Uniformly S-Multiplication Modules and Rings
by Wei Qi and Xiaolei Zhang
Mathematics 2023, 11(9), 2168; https://doi.org/10.3390/math11092168 - 5 May 2023
Viewed by 1349
Abstract
In this article, we introduce and study the notions of uniformly S-multiplication modules and rings that are generalizations of multiplication modules and rings. Some examples are given to distinguish the new conceptions with the old classical ones. Full article
12 pages, 276 KiB  
Article
Extension of Almost Primary Ideals to Noncommutative Rings and the Generalization of Nilary Ideals
by Alaa Abouhalaka and Şehmus Fındık
Mathematics 2023, 11(8), 1917; https://doi.org/10.3390/math11081917 - 18 Apr 2023
Cited by 4 | Viewed by 1515
Abstract
In this paper, we introduce the concepts of almost right primary ideals and almost nilary ideals and study their related results. We compare almost right primary ideals with other types of ideals, such as right primary ideals and weakly right primary ideals, and [...] Read more.
In this paper, we introduce the concepts of almost right primary ideals and almost nilary ideals and study their related results. We compare almost right primary ideals with other types of ideals, such as right primary ideals and weakly right primary ideals, and investigate their forms in decomposable rings. Moreover, we study the prime radical of an ideal of the product rings. Finally, we provide a definition of fully almost right primary rings and demonstrate that the homomorphic image of a fully almost right primary ring is again a fully almost right primary ring. We also investigate the quotient structure of fully almost right primary rings. Full article

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3 pages, 201 KiB  
Correction
Correction: Camazón Portela, D.; López Ramos, J.A. Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties. Mathematics 2023, 11, 3079
by Daniel Camazón Portela and Juan Antonio López Ramos
Mathematics 2024, 12(23), 3634; https://doi.org/10.3390/math12233634 - 21 Nov 2024
Viewed by 210
Abstract
In the published publication [...] Full article
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