Research

30 pages, 383 KiB

Open AccessFeature PaperArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math12081130 - 09 Apr 2024

Viewed by 350

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

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

20 pages, 847 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math11234770 - 25 Nov 2023

Viewed by 598

Abstract

Let m and n be fixed positive integers. Suppose that

A

is a von Neumann algebra with no central summands of type

I_{1}

, and

L_{m} : A \to A

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

I_{1}

, and

L_{m} : A \to A

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

ϕ_{m} : A \to A

and an additive higher map

ζ_{m} : A \to Z (A)

, which annihilates every

{(n - 1)}^{t h}

commutator

p_{n} (S_{1}, S_{2}, \dots, S_{n})

with

S_{1} S_{2} = 0

such that

L_{m} (S) = ϕ_{m} (S) + ζ_{m} (S) f o r a l l S \in A .

We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results. Full article

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

12 pages, 279 KiB

Open AccessArticle

Nonlinear Skew Lie-Type Derivations on ∗-Algebra

by Md Arshad Madni, Amal S. Alali and Muzibur Rahman Mozumder

Mathematics 2023, 11(18), 3819; https://0-doi-org.brum.beds.ac.uk/10.3390/math11183819 - 06 Sep 2023

Viewed by 794

Abstract

Let

A

be a unital ∗-algebra over the complex fields

C

. For any

H_{1}, H_{2} \in A

, a product

{[H_{1}, H_{2}]}_{•} = H_{1} H_{2} - H_{2} H_{1}^{*}

[...] Read more.

Let

A

be a unital ∗-algebra over the complex fields

C

. For any

H_{1}, H_{2} \in A

, a product

{[H_{1}, H_{2}]}_{•} = H_{1} H_{2} - H_{2} H_{1}^{*}

is called the skew Lie product. In this article, it is shown that if a map

ξ

:

A \to A

(not necessarily linear) satisfies

ξ (P_{n} (H_{1}, H_{2}, \dots, H_{n})) = \sum_{i = 1}^{n} P_{n} (H_{1}, \dots,

H_{i - 1}, ξ (H_{i}), H_{i + 1}, \dots, H_{n}) (n \geq 3)

for all

H_{1}, H_{2}, \dots, H_{n} \in A

, then

ξ

is additive. Moreover, if

ξ (i \frac{e}{2})

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

I_{1}

. Full article

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

22 pages, 353 KiB

Open AccessArticle

Acyclic Complexes and Graded Algebras

by Chaoyuan Zhou

Mathematics 2023, 11(14), 3167; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143167 - 19 Jul 2023

Viewed by 631

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

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

20 pages, 363 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math11143117 - 14 Jul 2023

Viewed by 664

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

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

12 pages, 293 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math11143079 - 12 Jul 2023

Viewed by 667

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

A_{2}

and

^{2} A_{4}

are given. Full article

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

8 pages, 272 KiB

Open AccessArticle

On Uniformly S-Multiplication Modules and Rings

by Wei Qi and Xiaolei Zhang

Mathematics 2023, 11(9), 2168; https://0-doi-org.brum.beds.ac.uk/10.3390/math11092168 - 05 May 2023

Viewed by 987

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

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

12 pages, 276 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math11081917 - 18 Apr 2023

Cited by 2 | Viewed by 1122

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

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

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