Let [Formula: see text] be a fixed integer. The purpose of this study is to explore [Formula: see text]-derivations on posets; we also provide characterization theorems for [Formula: see text]-derivations, analyze properties of the fixed points under these [Formula: see text]-derivations, and investigate properties of ideals and operations associated with them.

Abstract We associate a family of ideal sheaves to any ℚ-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and 𝑉-filtrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, provide criteria for the nontriviality, and introduce the center of minimal exponent (generalizing the notion of minimal log canonical center). The main idea is to exploit the global structure of the 𝑉-filtration along an effective divisor using the notion of twisted Hodge modules. As applications, we prove new cases of conjectures by Debarre, Casalaina–Martin, and Grushevsky on singularities of theta divisors on principally polarized abelian varieties.

This paper proposes an extended framework for bipolar anti-intuitionistic fuzzy ideals within the context of ordered Γ-semigroups. We introduce and investigate the (δ, τ)-bipolar anti-intuitionistic fuzzy subsemigroups (BPAIFSS), including their associated left ideals, right ideals, ideals, and bi-ideals. These structures generalize existing fuzzy ideal notions by incorporating dual-valued membership and non-membership functions with flexible threshold control. Using level set analysis, we characterize the algebraic properties of these fuzzy ideals and establish their role in determining the regularity of ordered Γ-semigroups. Illustrative examples are provided to validate and demonstrate the applicability of the theoretical results.

Abstract This paper presents a comprehensive study of right $${\textbf {S}}$$ S -maximal ideals in noncommutative rings, providing a natural extension of classical ideal theory through the framework of $${\textbf {m}}$$ m -systems. We establish deep connections among right $${\textbf {S}}$$ S -maximal, S -comaximal, and $${\textbf {S}}$$ S -prime ideals, and introduce new concepts such as the $${\textbf {S}}$$ S -Jacobson radical and $${\textbf {S}}$$ S -invertible elements to further develop the structural theory of rings. A key contribution is the introduction of $${\textbf {S}}$$ S -local rings and the formulation of an $${\textbf {S}}$$ S -version of Nakayama’s Lemma. In addition, we propose two complementary definitions of left $${\textbf {S}}$$ S -primitive ideals–one ideal-theoretic and the other annihilator-based–and demonstrate their intrinsic relationship to $${\textbf {S}}$$ S -prime ideals. Our results not only unify and generalize existing notions but also open promising avenues for further research, particularly in exploring the interplay between different forms of $${\textbf {S}}$$ S -primitivity.

Classical mathematical methods are insufficient for resolving certain issuesin real-life human problems due to the uncertainty of the data. Researchers from around the world have created innovative mathematical models, like soft and fuzzy set theories, to model the uncertainties that arise in different areas. Jun recently developed a hybrid structure that combined fuzzy and soft set concepts. The hybrid structure principle is appliedto groupoids in this paper, and the properties of hybrid ideals and hybrid subgroupoids in groupoids are also described. Furthermore, the notions of hybrid subgroups, hybrid normal subgroups, and hybrid cosets in a group, as well as their key properties, are discussed. In addition, we show that any member of the collection of hybrid cut sets of a hybrid normal subgroup of a group G is a normal subgroup of G in the traditional sense. Finally, we obtain a finite-group hybrid version of Lagrange’s theorem.

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.

This paper aims primarily to identify necessary and sufficient conditions under which the Rees algebra R[IX] of an ideal I in a commutative ring R is a McCoy ring. For this sake, we define a McCoy ideal as an ideal J such that ann R ( J ) is a McCoy R-module. We investigate the properties of such ideals and provide several illustrative examples. One of our key results proves that if I is a finitely generated ideal of R, then R[IX] is a McCoy ring if and only if I is a McCoy ideal. Additionally, we revisit a result from [16] which asserts that every nontrivial graded ring is a McCoy ring [16, Theorem 2.7]. Our study explores the McCoyness of Rees algebras R [ IX ] = ⊕ n ∈ N I n X n offering a novel approach to understanding trivial graded rings and highlighting how their McCoy property depends on the characteristics of the considered ideal I.

Abstract We give characterizations for the thin ideal and almost thin ideal, which were studied by Jörg Brendle and Jana Flašková in their paper Brendle, J. 236 (3), 201–245 (2017). We check the various properties of selected ideals. We also verify whether these ideals are comparable in the Rudin-Blass order and Rudin-Keisler order. Moreover, we present a negative answer to [Filipów, R. 1–48 (2024) Question 8.8(1)].

Abstract We study ideals and filters of posets and of pseudocomplemented posets and show a version of the Separation Theorem, known for ideals and filters in lattices and semilattices, within this general setting. We extend the concept of a $$*$$ -ideal already introduced by Rao for pseudocomplemented distributive lattices and by Talukder, Chakraborty and Begum for pseudocomplemented semilattices to pseudocomplemented posets. We derive several important properties of such ideals. Especially, we explain connections between prime filters, ultrafilters, filters satisfying the $$*$$ -condition and dense elements. Finally, we prove a Separation Theorem for $$*$$ -ideals.

Let [Formula: see text] denote the class of closed graphs with Cohen-Macaulay binomial edge ideals and [Formula: see text] denote the class of proper interval graphs. Then [Formula: see text] The [Formula: see text]-completion problem is a classical problem in graph theory as well as in molecular biology, and this problem is known to be NP-hard. In this paper, we study the [Formula: see text]-completion problem. We give a method to construct all possible [Formula: see text]-completions of a graph. We find the [Formula: see text]-completion number and the set of all minimal [Formula: see text]-completions for a large class of graphs. Moreover, for this class, we give a polynomial-time algorithm to compute the [Formula: see text]-completion number and a minimum [Formula: see text]-completion of a given graph. The unmixedness and Cohen-Macaulay properties of binomial edge ideals of induced subgraphs are investigated. Also, we discuss the accessible graph completion and the Cohen-Macaulay property of binomial edge ideals of whisker graphs.

Let [Formula: see text] be an ideal in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] has the copersistence property if [Formula: see text] for all [Formula: see text]. The first aim of this paper is to explore this concept for monomial ideals thus we introduce some classes of monomial ideals satisfying this definition and also studying this notion under monomial operations. In addition, we say that [Formula: see text] has nearly copersistence property if there exist a positive integer [Formula: see text] and a monomial prime ideal [Formula: see text] such that [Formula: see text] for all [Formula: see text], and [Formula: see text] for all [Formula: see text]. The second goal of this paper is to investigate monomial ideals satisfying this concept.

In this paper, we define essential fuzzy ideals of semirings and investigate some properties of them. We give some properties of essential ideals and essential fuzzy ideals. Moreover, we show relationships between essential ideals and essential fuzzy ideals.

This article is devoted to the investigation of some algebraic connections between algebraic structures and algebraic hyperstructures. Firstly, we introduce the concept of ordered power ternary semigroups on ternary semihypergroups induced by posets which are generalizations of power ternary semigroups on ternary semihypergroups. Then, the ideas of ideals and pure ideals in ordered power ternary semigroups on ternary semihypergroups induced by posets are introduced. We also study some algebraic properties of pure ideals and weakly pure ideals on the ordered ternary semigroups.

Abstract Recently we have presented a unified approach to two classes of Banach spaces defined by means of variations (Waterman spaces and Chanturia classes), utilizing the concepts from the theory of ideals on the set of natural numbers. We defined correspondence between an ideal on the set of natural numbers, a certain sequence space and related space of functions of bounded variation. In this paper, following these ideas, we give characterizations of compact embeddings between different Waterman spaces and between different Chanturia classes: both in terms of sequences defining these function spaces and in terms of properties of ideals corresponding to these function spaces.

Let [Formula: see text] be a commutative ring and [Formula: see text] an endomorphism of [Formula: see text]. In this paper we introduce and study the concept of [Formula: see text]-primary ideal which is a generalization of primary ideal. An ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-primary if whenever [Formula: see text] for some [Formula: see text] then [Formula: see text] or [Formula: see text]. We show that [Formula: see text]-primary ideals share many analogous properties of primary ideals and we also describe the behavior of [Formula: see text]-primary property across various ring theoretic constructions such as trivial ring extensions and amalgamations.

The complex intuitionistic fuzzy set is an extension of the intuitionistic fuzzy set where the membership and non-membership functions are expressed by a complex numbers. Ring theory is a well-known field of abstract algebra that is used in a broad area of present study in mathematics and computer science. The study of ideals is important in numerous ways in ring theory. Keeping in view the importance of complex intuitionistic fuzzy sets and ring theory, in this paper, we define the notion of complex intuitionistic fuzzy ideals in a classical ring R and investigate its various algebraic properties. We obtain that the intersection of any two complex intuitionistic fuzzy ideals of a classical ring R is again a complex intuitionistic fuzzy ideal of R. We also define the notion of a complex intuitionistic fuzzy level set. Furthermore, we define the concept of complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal of a classical ring and prove that the set of all complex intuitionistic fuzzy cosets of a complex intuitionistic fuzzy ideal forms a ring under certain binary operations. Finally, we prove a complex intuitionistic fuzzy version of the fundamental theorem of a ring homomorphism.

The fuzzy set is an excellent tool for dealing with indeterminacy that can be clearly and effectively analysed from the decision-maker’s viewpoint, and it is extremely helpful for showing people’s hesitations in their everyday interactions. In order to deal with practical problems, soft set theory has recently been created. By combining the fuzzy and soft sets, Jun et al. [Hybrid structures and applications, Annals of Communications in Mathematics 1(1) (2018) 11–25] created hybrid structures. Hybrid structures are a combination of soft-set and fuzzy-set theories. The concepts of hybrid bi-interior ideals of semigroups are addressed in this paper. We also characterize regular semigroups in terms of hybrid bi-interior ideals of semigroups.

This paper introduces the concept of right S-semiprime ideals and right S-semiprime submodules in the context of noncommutative rings. We investigate the properties of right S-semiprime ideals (submodules), providing multiple equivalent definitions and exploring their relationship with classical semiprime ideals (submodules). Key results include the definitions and characterizations of S-radical and S-pseudo radical concepts for ideals. The paper concludes with new insights into the structure of right S-semiprime submodules.

Abstract We study the behavior of the Hilbert–Kunz multiplicity of powers of an ideal in a local ring. In dimension 2, we provide answers to some problems raised by Smirnov, and give a criterion to answer one of his questions in terms of a “Ratliff–Rush version” of the Hilbert–Kunz multiplicity.

Introduction: We propose the -norm of the Anti-fuzzy T-subalgebra and the T-ideal of the BP-algebra, and investigate some of their properties in this work. In addition, we define properties of Cartesian products of Anti fuzzy T subalgebras and T-ideals of BP algebras. These are treated in detail along with other algebraic properties. Objectives: The primary objective of this study is to investigate the combinatorial properties of Anti T-subalgebras and T-ideals within the framework of fuzzy BP-algebras using ᵵ-norms. The research aims to explore and characterize their algebraic structures, including operations, interconnections, intersections, and Cartesian products. Methods: The concepts of t-norms can be applied to explore the algebraic properties of anti-fuzzy T-ideals and T-subalgebras within BP-algebras. These concepts are particularly relevant in formulating and proving theorems, lemmas, and illustrative examples related to the structure and behavior of such algebraic systems. Conclusions: In this study, we investigated the ȶ-norm of Anti-fuzzy T-subalgebras and T-ideals within the structure of BP-algebras and established several fundamental properties. We further introduced the concept of Cartesian products of Anti-fuzzy T-subalgebras and T-ideals, demonstrating key structural characteristics and interactions. The interconnection and intersection of these algebraic structures were also analyzed in depth, offering new insights into their theoretical behavior. The findings presented in this work provide a solid foundation for future extensions and generalizations. In particular, the notions developed here can be further explored in the context of intuitionistic Q-fuzzy sets, interval-valued Q-fuzzy sets, and Q-bipolar fuzzy sets, opening avenues for advanced research in generalized fuzzy algebraic systems.