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13 pages, 222 KB

Open AccessArticle

Notes on Semiprime Ideals with Symmetric Bi-Derivation

by Ali Yahya Hummdi, Öznur Gölbaşı, Emine Koç Sögütcü and Nadeem ur Rehman

Axioms 2025, 14(4), 260; https://doi.org/10.3390/axioms14040260 - 29 Mar 2025

Viewed by 353

In this paper, we prove many algebraic identities that include symmetric bi-derivation in rings which contain a semiprime ideal. We intend to generalize previous results obtained for semiprime rings with symmetric derivation using semiprime ideals in rings. Full article

(This article belongs to the Special Issue Advances in Applied Algebra and Related Topics)

13 pages, 235 KB

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 703

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)

18 pages, 604 KB

Open AccessArticle

Exploring the Structure of Possibility Multi-Fuzzy Soft Ordered Semigroups Through Interior Ideals

by Sana Habib, Kashif Habib, Violeta Leoreanu-Fotea and Faiz Muhammad Khan

Mathematics 2025, 13(2), 210; https://doi.org/10.3390/math13020210 - 9 Jan 2025

Viewed by 758

Abstract

This paper aims to introduce a novel idea of possibility multi-fuzzy soft ordered semigroups for ideals and interior ideals. Various results, formulated as theorems based on these concepts, are presented and further validated with suitable examples. This paper also explores the broad applicability [...] Read more.

This paper aims to introduce a novel idea of possibility multi-fuzzy soft ordered semigroups for ideals and interior ideals. Various results, formulated as theorems based on these concepts, are presented and further validated with suitable examples. This paper also explores the broad applicability of possibility multi-fuzzy soft ordered semigroups in solving modern decision-making problems. Furthermore, this paper explores various classes of ordered semigroups, such as simple, regular, and intra-regular, using this innovative method. Based on these concepts, some important conclusions are drawn with supporting examples. Moreover, it defines the possibility of multi-fuzzy soft ideals for semiprime ordered semigroups. Full article

(This article belongs to the Special Issue Fuzzy Logic and Soft Computing—In Memory of Lotfi A. Zadeh)

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12 pages, 257 KB

Open AccessArticle

Factor Rings with Algebraic Identities via Generalized Derivations

by Ali Yahya Hummdi, Zakia Z. Al-Amery and Radwan M. Al-omary

Axioms 2025, 14(1), 15; https://doi.org/10.3390/axioms14010015 - 30 Dec 2024

Cited by 4 | Viewed by 651

Abstract

The current article focuses on studying the behavior of a ring

ℜ / Π

when ℜ admits generalized derivations

Ψ

and

Ω

with associated derivations

ϕ

and

δ

, respectively. These derivations satisfy specific differential identities involving

Π

, where

Π

is a [...] Read more.

The current article focuses on studying the behavior of a ring

ℜ / Π

when ℜ admits generalized derivations

Ψ

and

Ω

with associated derivations

ϕ

and

δ

, respectively. These derivations satisfy specific differential identities involving

Π

, where

Π

is a prime ideal of an arbitrary ring ℜ, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of

Π

in the hypotheses of our various theorems, we provide a list of examples. Full article

(This article belongs to the Section Algebra and Number Theory)

15 pages, 270 KB

Open AccessArticle

Symmetric Reverse n-Derivations on Ideals of Semiprime Rings

by Shakir Ali, Ali Yahya Hummdi, Naira N. Rafiquee, Vaishali Varshney and Kok Bin Wong

Axioms 2024, 13(10), 717; https://doi.org/10.3390/axioms13100717 - 16 Oct 2024

Cited by 1 | Viewed by 843

Abstract

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. [...] Read more.

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen our understanding of these symmetric maps and their properties within mathematical structures. Full article

(This article belongs to the Section Algebra and Number Theory)

14 pages, 244 KB

Open AccessArticle

Some Identities Related to Semiprime Ideal of Rings with Multiplicative Generalized Derivations

by Ali Yahya Hummdi, Emine Koç Sögütcü, Öznur Gölbaşı and Nadeem ur Rehman

Axioms 2024, 13(10), 669; https://doi.org/10.3390/axioms13100669 - 27 Sep 2024

Viewed by 878

Abstract

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let

F

be a ring with a semiprime ideal

Π

. A map

ϕ : F \to F

is classified as a multiplicative generalized derivation [...] Read more.

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let

F

be a ring with a semiprime ideal

Π

. A map

ϕ : F \to F

is classified as a multiplicative generalized derivation if there exists a map

σ : F \to F

such that

ϕ (x y) = ϕ (x) y + x σ (y)

for all

x, y \in F

. This study focuses on semiprime ideals

Π

that admit multiplicative generalized derivations

ϕ

and G that satisfy certain differential identities within

F

. By examining these conditions, the paper aims to provide new insights into the structural aspects of rings, particularly their commutativity in relation to the behavior of such derivations. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

13 pages, 228 KB

Open AccessArticle

Some Equations in Rings Involving Semiprime Ideals and Multiplicative Generalized Semiderivations

by Ali Yahya Hummdi, Öznur Gölbaşı, Emine Koç Sögütcü and Nadeem ur Rehman

Mathematics 2024, 12(18), 2818; https://doi.org/10.3390/math12182818 - 11 Sep 2024

Viewed by 969

Abstract

This paper examines the commutativity of the quotient ring

F / Y

by utilizing specific differential identities in a general ring

F

that contains a semiprime ideal

Y

. This study particularly focuses on the role of a multiplicative generalized semiderivation

ψ

, [...] Read more.

This paper examines the commutativity of the quotient ring

F / Y

by utilizing specific differential identities in a general ring

F

that contains a semiprime ideal

Y

. This study particularly focuses on the role of a multiplicative generalized semiderivation

ψ

, which is associated with a map

θ

, in determining the commutative nature of the quotient ring. Full article

13 pages, 252 KB

Open AccessArticle

Center-like Subsets in Semiprime Rings with Multiplicative Derivations

by Sarah Samah Aljohani, Emine Koç Sögütcü and Nadeem ur Rehman

Axioms 2024, 13(7), 448; https://doi.org/10.3390/axioms13070448 - 2 Jul 2024

Cited by 1 | Viewed by 922

Abstract

We introduce center-like subsets

Z_{\circ}^{*} (A, d), Z_{\circ}^{* *} (A, d),

where

A

is the ring and

d

is the multiplicative derivation. In the following, we take a new derivation for [...] Read more.

We introduce center-like subsets

Z_{\circ}^{*} (A, d), Z_{\circ}^{* *} (A, d),

where

A

is the ring and

d

is the multiplicative derivation. In the following, we take a new derivation for the center-like subsets existing in the literature and establish the relations between these sets. In addition to these new sets, the theorems are generalized as multiplicative derivations instead of the derivations found in previous studies. Additionally, different proofs are provided for different center-like sets. Finally, we enrich this article with examples demonstrating that the hypotheses we use are necessary. Full article

24 pages, 444 KB

Open AccessArticle

Exploring Hybrid H-bi-Ideals in Hemirings: Characterizations and Applications in Decision Making

by Asmat Hadi, Asghar Khan, Nosheen Faiz, Dost Muhammad Khan, Rashad A. R. Bantan and Mohammed Elgarhy

Mathematics 2023, 11(17), 3683; https://doi.org/10.3390/math11173683 - 26 Aug 2023

Viewed by 1368

Abstract

The concept of the hybrid structure, as an extension of both soft sets and fuzzy sets, has gained significant attention in various mathematical and decision-making domains. In this paper, we delve into the realm of hemirings and investigate the properties of hybrid h-bi-ideals, [...] Read more.

The concept of the hybrid structure, as an extension of both soft sets and fuzzy sets, has gained significant attention in various mathematical and decision-making domains. In this paper, we delve into the realm of hemirings and investigate the properties of hybrid h-bi-ideals, including prime, strongly prime, semiprime, irreducible, and strongly irreducible ones. By employing these hybrid h-bi-ideals, we provide insightful characterizations of h-hemiregular and h-intra-hemiregular hemirings, offering a deeper understanding of their algebraic structures. Beyond theoretical implications, we demonstrate the practical value of hybrid structures and decision-making theory in handling real-world problems under imprecise environments. Using the proposed decision-making algorithm based on hybrid structures, we have successfully addressed a significant real-world problem, showcasing the efficacy of this approach in providing robust solutions. Full article

(This article belongs to the Special Issue Fuzzy Decision Making and Applications)

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20 pages, 363 KB

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://doi.org/10.3390/math11143117 - 14 Jul 2023

Cited by 4 | Viewed by 1490

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)

22 pages, 692 KB

Open AccessArticle

A Detailed Study of Mathematical Rings in q-Rung Orthopair Fuzzy Framework

by Asima Razzaque, Abdul Razaq, Ghaliah Alhamzi, Harish Garg and Muhammad Iftikhar Faraz

Symmetry 2023, 15(3), 697; https://doi.org/10.3390/sym15030697 - 10 Mar 2023

Cited by 7 | Viewed by 2275

Abstract

Symmetry-related problems can be addressed by means of group theory, and ring theory can be seen as an extension of additive group theory. Ring theory, a significant topic in abstract algebra, is currently active in a diverse range of study domains across the [...] Read more.

Symmetry-related problems can be addressed by means of group theory, and ring theory can be seen as an extension of additive group theory. Ring theory, a significant topic in abstract algebra, is currently active in a diverse range of study domains across the disciplines of mathematics, theoretical physics and coding theory. The study of ideals is vital to the theory of rings in a wide range of ways. The uncertainties present in the information are addressed well by the q-rung orthopair fuzzy set (q-ROFS). Considering the significance of ring theory and the q-ROFS, this article defines q-rung orthopair fuzzy ideals (q-ROFIs) in conventional rings and investigates its various algebraic features. We introduce the notion of q-rung orthopair fuzzy cosets (q-ROFCs) of a q-ROFI and demonstrate that, under certain binary operations, the collection of all q-ROFCs of a q-ROFI forms a ring. In addition, we provide a q-rung orthopair analog of the fundamental theorem of ring homomorphism. Furthermore, we present the notion of q-rung orthopair fuzzy semi-prime ideals (q-ROFSPIs) and provide a comprehensive explanation of their many algebraic properties. Finally, regular rings were characterized using q-ROFIs. Full article

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16 pages, 303 KB

Open AccessArticle

Hybrid Nil Radical of a Ring

by Kasi Porselvi, Ghulam Muhiuddin, Balasubramanian Elavarasan and Abdullah Assiry

Symmetry 2022, 14(7), 1367; https://doi.org/10.3390/sym14071367 - 3 Jul 2022

Cited by 17 | Viewed by 2044

Abstract

The nature of universe problems is ambiguous due to the presence of asymmetric data in almost all disciplines, including engineering, mathematics, medical sciences, physics, computer science, operations research, artificial intelligence, and management sciences, and they involve various types of uncertainties when dealing with [...] Read more.

The nature of universe problems is ambiguous due to the presence of asymmetric data in almost all disciplines, including engineering, mathematics, medical sciences, physics, computer science, operations research, artificial intelligence, and management sciences, and they involve various types of uncertainties when dealing with them on various occasions. To deal with the challenges of uncertainty and asymmetric information, different theories have been developed, including probability, fuzzy sets, rough sets, soft ideals, etc. The strategies of hybrid ideals, hybrid nil radicals, hybrid semiprime ideals, and hybrid products of rings are introduced in this paper and hybrid structures are used to examine the structural properties of rings. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

3 pages, 205 KB

Open AccessArticle

On Rings of Weak Global Dimension at Most One

by Askar Tuganbaev

Mathematics 2021, 9(21), 2643; https://doi.org/10.3390/math9212643 - 20 Oct 2021

Cited by 2 | Viewed by 1933

Abstract

A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals [...] Read more.

A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of R is distributive. Jensen has proved earlier that a commutative ring R is a ring of weak global dimension at most one if and only if R is an arithmetical semiprime ring. A ring R is said to be centrally essential if either R is commutative or, for every noncentral element

x \in R

, there exist two nonzero central elements

y, z \in R

with

x y = z

. In Theorem 2 of our paper, we prove that a centrally essential ring R is of weak global dimension at most one if and only is R is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings. Full article

(This article belongs to the Special Issue Modules over Noncommutative Rings, Homological Algebra and Noncommutative Geometry)

9 pages, 276 KB

Open AccessArticle

Rings with Boolean Lattices of One-Sided Annihilators

by Małgorzata Jastrzębska

Symmetry 2021, 13(10), 1909; https://doi.org/10.3390/sym13101909 - 11 Oct 2021

Cited by 2 | Viewed by 1988

Abstract

The present paper is part of the research on the description of rings with a given property of the lattice of left (right) annihilators. The anti-isomorphism of lattices of left and right annihilators in any ring gives some kind of symmetry: the lattice [...] Read more.

The present paper is part of the research on the description of rings with a given property of the lattice of left (right) annihilators. The anti-isomorphism of lattices of left and right annihilators in any ring gives some kind of symmetry: the lattice of left annihilators is Boolean (complemented, distributive) if and only if the lattice of right annihilators is such. This allows us to restrict our investigations mainly to the left side. For a unital associative ring R, we prove that the lattice of left annihilators in R is Boolean if and only if R is a reduced ring. We also prove that the lattice of left annihilators of R being two-sided ideals is complemented if and only if this lattice is Boolean. The last statement, in turn, is known to be equivalent to the semiprimeness of

R .

On the other hand, for any complete lattice L, we construct a nilpotent ring whose lattice of left annihilators coincides with its sublattice of left annihilators being two-sided ideals and is isomorphic to

L .

This construction shows that the assumption of R being unital cannot be dropped in any of the above two results. Some additional results on rings with distributive or complemented lattices of left annihilators are obtained. Full article

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10 pages, 278 KB

Open AccessArticle

On Graded 2-Prime Ideals

by Malik Bataineh and Rashid Abu-Dawwas

Mathematics 2021, 9(5), 493; https://doi.org/10.3390/math9050493 - 28 Feb 2021

Cited by 8 | Viewed by 2347

Abstract

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ideals and graded [...] Read more.

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ideals and graded weakly prime ideals are also different. Several properties of graded 2-prime ideals are investigated. We study graded rings in which every graded 2-prime ideal is graded prime, we call such a graded ring a graded 2-P-ring. Moreover, we introduce the concept of graded semi-primary ideals, and show that graded 2-prime ideals and graded semi-primary ideals are different concepts. In fact, we show that graded semi-primary, graded 2-prime and graded primary ideals are equivalent over

Z

-graded principal ideal domain. Full article

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