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12 pages, 247 KiB

Open AccessArticle

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 395

Abstract

In this article, for a prime ideal

ρ

of an arbitrary ring ℜ, we study the commutativity of the quotient ring

ℜ / 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

ℜ / 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

Open AccessArticle

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 541

Abstract

Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of

S .

An additive mapping ℏ 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 ℏ on S is defined as a homoderivation if

ℏ (a b) = ℏ (a) ℏ (b) + ℏ (a) b + a ℏ (a)

for all

a, b \in S .

In the present paper, we shall prove that ℏ is a commuting map on U if any one of the following holds: (i)

ℏ ({\tilde{a}}_{1} {\tilde{a}}_{2}) + {\tilde{a}}_{1} {\tilde{a}}_{2} \in Z,

(ii)

ℏ ({\tilde{a}}_{1} {\tilde{a}}_{2}) - {\tilde{a}}_{1} {\tilde{a}}_{2} \in Z,

(iii)

ℏ ({\tilde{a}}_{1} \circ {\tilde{a}}_{2}) = 0,

(iv)

ℏ ({\tilde{a}}_{1} \circ {\tilde{a}}_{2}) = [{\tilde{a}}_{1}, {\tilde{a}}_{2}],

(v)

ℏ ([{\tilde{a}}_{1}, {\tilde{a}}_{2}]) = 0,

(vi)

ℏ ([{\tilde{a}}_{1}, {\tilde{a}}_{2}]) =

({\tilde{a}}_{1} \circ {\tilde{a}}_{2}),

(vii)

{\tilde{a}}_{1} ℏ ({\tilde{a}}_{2}) \pm {\tilde{a}}_{1} {\tilde{a}}_{2} \in Z,

(viii)

{\tilde{a}}_{1} ℏ ({\tilde{a}}_{2}) \pm {\tilde{a}}_{2} {\tilde{a}}_{1} = 0,

(ix)

{\tilde{a}}_{1} ℏ ({\tilde{a}}_{2}) \pm {\tilde{a}}_{1} \circ {\tilde{a}}_{2} = 0,

(x)

[ℏ ({\tilde{a}}_{1}), {\tilde{a}}_{2}] \pm {\tilde{a}}_{1} {\tilde{a}}_{2} = 0,

(xi)

[ℏ ({\tilde{a}}_{1}), {\tilde{a}}_{2}] \pm {\tilde{a}}_{2} {\tilde{a}}_{1} = 0,

for all

{\tilde{a}}_{1}, {\tilde{a}}_{2} \in U,

where ℏ 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

Open AccessArticle

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 486

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

Open AccessArticle

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

Cited by 1 | Viewed by 769

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

a b + c d

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

a b + c d

is a nontrivial idempotent, then

b a + d c

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

Open AccessArticle

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 624

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

R_{e_{G}}

-module. Moreover, we prove that the additive induction functor

{(-)}^{R}

and the restriction functor

{(-)}_{e_{G}}

form an equivalence between the categories

G - GrR - Mod

and

R_{e_{G}} - 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

Open AccessArticle

The Connection between Der(

U_{q}^{+} (g)

) and Der(

U_{r, s}^{+} (g)

)

by Yongyue Zhong and Xiaomin Tang

Mathematics 2024, 12(16), 2517; https://doi.org/10.3390/math12162517 - 15 Aug 2024

Viewed by 643

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

Open AccessArticle

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 4 | Viewed by 864

Abstract

The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring

ℑ / ℘

, where ℘ is a prime ideal in a ring ℑ. 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

ℑ / ℘

, where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized

(α, β)

-derivation associated with an

(α, β)

-derivation ∂ that satisfies certain criteria involving ℘, where

α

and

β

are automorphisms on ℑ. 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

Open AccessArticle

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 1067

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

Open AccessArticle

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 926

Abstract

Let s, t be two positive integers and

k

be an algebraically closed field with char (

k) ∤ s t

. We show that the Drinfeld double

D (⋀_{s t, t}^{* c o p})

of [...] Read more.

Let s, t be two positive integers and

k

be an algebraically closed field with char (

k) ∤ s t

. We show that the Drinfeld double

D (⋀_{s t, t}^{* c o p})

of generalized Taft–Hopf algebra

⋀_{s t, t}^{* c o p}

has ribbon elements if and only if t is odd. Moreover, if s is even and t is odd, then

D (⋀_{s t, t}^{* c o p})

has two ribbon elements, and if both s and t are odd, then

D (⋀_{s t, t}^{* c o p})

has only one ribbon element. Finally, we compute explicitly all ribbon elements of

D (⋀_{s t, t}^{* c o p})

. Full article

(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra)

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