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11 pages, 275 KB

Open AccessArticle

On Axis-Reversible Rings

by Muhammad Saad and Majed Zailaee

Mathematics 2026, 14(3), 492; https://doi.org/10.3390/math14030492 - 30 Jan 2026

Viewed by 330

This work explores the notion of axis-reversible rings, a generalization of axis-commutative rings. The objective is to investigate their characteristics and relevance within the wider context of ring theory. This paper defines axis-reversibility and demonstrates its importance through many examples. It also analyzes the characteristics of several matrix rings, elucidating the conditions under which a ring can be deemed axis-reversible. This paper examines the relationship between axis-reversibility and other significant ring qualities, such as reducedness and semiprimeness, through comprehensive arguments and proofs. This study provides novel perspectives on non-commutative rings, enhancing our comprehension of algebraic structures. Full article

(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra, 2nd Edition)

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17 pages, 337 KB

Open AccessArticle

Rings Whose Nilpotent Elements Form a Wedderburn Radical Subring

by Ryszard Mazurek

Symmetry 2025, 17(11), 1815; https://doi.org/10.3390/sym17111815 - 28 Oct 2025

Viewed by 591

Abstract

We introduce and study a class of rings, which we call NRW rings, distinguished by the condition that the set of nilpotent elements forms a Wedderburn radical subring. This class includes symmetric and semicommutative rings, as well as many other well-known and important classes. We also define nilpotent-semicommuting rings as those satisfying the semicommutativity condition restricted to nilpotent elements and prove that every nilpotent-semicommuting ring is an NRW ring. We provide an element-wise characterization of NRW rings and show that the NRW property is symmetric with respect to one-sided principal ideals. Based on the right ideals of a ring that are NRW, for any ring R we inductively construct the ideal

\bar{W} (R)

and prove that the prime radical of R equals the intersection of

\bar{W} (R)

with the sum of all nil right ideals of R. As a consequence, a positive solution to the Köthe conjecture follows for all rings R satisfying

R = \bar{W} (R)

. We also characterize when certain classical ring constructions, such as direct sums, matrix rings, and the Dorroh extension, yield an NRW ring. Full article

(This article belongs to the Section Mathematics)

13 pages, 262 KB

Open AccessArticle

Semicommutative and Armendariz Matrix Rings

by Gang Yang

Axioms 2025, 14(11), 787; https://doi.org/10.3390/axioms14110787 - 26 Oct 2025

Viewed by 461

Abstract

In this paper, we construct some interesting high-order upper triangular matrix rings, which have semicommutative and Armendariz properties. Also, the relatively maximality of these rings as subrings of certain matrix rings is considered. Full article

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

12 pages, 233 KB

Open AccessArticle

Dual Toeplitz Operators on Bounded Symmetric Domains

by Jianxiang Dong

Mathematics 2025, 13(10), 1611; https://doi.org/10.3390/math13101611 - 14 May 2025

Viewed by 565

Abstract

We give some characterizations of dual Toeplitz operators acting on the orthogonal complement of the Bergman space over bounded symmetric domains. Our main result characterizes those finite sums of products of Toeplitz operators that are themselves dual Toeplitz operators. Furthermore, we obtain a necessary condition for such finite sums of dual Toeplitz products to be compact. As an application of our main result, we derive a sufficient and necessary condition for when the (semi-)commutators of dual Toeplitz operators is zero. Notably, we find that a dual Toeplitz operator is compact if and only if it is the zero operator. Full article

11 pages, 212 KB

Open AccessArticle

Regularity of Idempotent Reflexive GP-V’-Rings

by Liuwen Li, Wenlin Zou and Ying Li

Mathematics 2024, 12(20), 3265; https://doi.org/10.3390/math12203265 - 17 Oct 2024

Cited by 1 | Viewed by 855

Abstract

This paper discusses the regularity of the GP-V’-rings in conjunction with idempotent reflexivity for the first time. We mainly discuss the weak and strong regularity of the GP-V’-rings using generalized weak ideals, weakly right ideals, and quasi-ideals. We show the following: (1) If [...] Read more.

R

is an idempotent reflexive semi-abelian left GP-V’-ring whose every maximal essential left ideal is a generalized weak ideal, a weakly right ideal, or a quasi-ideal, then

R

is a reduced left weakly regular ring. (2)

R

is a strongly regular ring if and only if

R

is an idempotent reflexive semi-commutative left GP-V’-ring whose every maximal essential left ideal is a generalized weak ideal, a weakly right ideal, or a quasi-ideal. (3) If

R

is a semi-primitive idempotent reflexive ring whose every simple singular left

R

-module is flat, and every maximal left ideal is a generalized weak ideal, then, for any nonzero element

a \in R

, there exists a positive integer

n

such that

a^{n} \neq 0

, and

R a R + l (a^{n}) = R

. Full article

13 pages, 267 KB

Open AccessArticle

On Nilpotent Elements and Armendariz Modules

by Nazeer Ansari, Kholood Alnefaie, Shakir Ali, Adnan Abbasi and Kh. Herachandra Singh

Mathematics 2024, 12(19), 3133; https://doi.org/10.3390/math12193133 - 7 Oct 2024

Cited by 1 | Viewed by 1332

Abstract

For a left module

{}_{R}M

over a non-commutative ring R, the notion for the class of nilpotent elements (

n i l_{R} (M)

) was first introduced and studied by Sevviiri and Groenewald in 2014 (Commun. Algebra [...] Read more.

For a left module

{}_{R}M

over a non-commutative ring R, the notion for the class of nilpotent elements (

n i l_{R} (M)

) was first introduced and studied by Sevviiri and Groenewald in 2014 (Commun. Algebra, 42, 571–577). Moreover, Armendariz and semicommutative modules are generalizations of reduced modules and

n i l_{R} (M) = 0

in the case of reduced modules. Thus, the nilpotent class plays a vital role in these modules. Motivated by this, we present the concept of nil-Armendariz modules as a generalization of reduced modules and a refinement of Armendariz modules, focusing on the class of nilpotent elements. Further, we demonstrate that the quotient module

M / N

is nil-Armendariz if and only if N is within the nilpotent class of

{}_{R}M

. Additionally, we establish that the matrix module

M_{n} (M)

is nil-Armendariz over

M_{n} (R)

and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization. Full article

48 pages, 454 KB

Open AccessArticle

New Types of Permuting n-Derivations with Their Applications on Associative Rings

by Mehsin Jabel Atteya

Symmetry 2020, 12(1), 46; https://doi.org/10.3390/sym12010046 - 25 Dec 2019

Cited by 5 | Viewed by 3016

Abstract

In this article, we introduce new generators of a permuting n-derivations to improve and increase the action of usual derivation. We produce a permuting n-generalized semiderivation, a permuting n-semigeneralized semiderivation, a permuting n-antisemigeneralized semiderivation and a permuting skew n-antisemigeneralized semiderivation of non-empty rings with their applications. Actually, we study the behaviour of those types and present their results of semiprime ring R. Examples of various results have also been included. That is, many of the branches of science such as business, engineering and quantum physics, which used a derivation, have the opportunity to invest them in solving their problems. Full article

(This article belongs to the Special Issue Commutative Ring Theory, Commutative Rings and Symmetry)

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