Properties Of Ideals Research Articles

On Lie Homomorphisms of Complex IntuitionisticFuzzy Lie Algebras

In this paper, we investigate the properties of complex intuitionistic fuzzy Lie subalgebras and ideals under Lie algebra homomorphisms, focusing on both images and inverse images. We provide detailed proofs for several new results concerning the preservation of complex intuitionistic fuzzy structures through homomorphisms, and we introduce additional homomorphism-related properties for these structures. This work extends known results to the context of complex intuitionistic fuzzy Lie algebras, contributing new insights into their behavior under algebraic mappings.

Properties of Hybrid Ideals of Rings

Properties of Hybrid Ideals of Rings

Higher Jacobian ideals, contact equivalence and motivic zeta functions

We show basic properties of higher Jacobian matrices and higher Jacobian ideals for functions and apply it to obtain two main results concerning singularities of functions. Firstly, we prove that a higher Nash blowup algebra is invariant under contact equivalences, which was recently conjectured by Hussain, Ma, Yau and Zuo. Secondly, we obtain an analogue of a result on motivic nearby cycles by Bussi, Joyce and Meinhardt.

A new algorithm for Gröbner bases conversion

A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering.The main ingredient of the new algorithm is the computation of a Gröbner basis with respect to the target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the ideal with respect to the target ordering. In general, more than one iteration may be needed to get a Gröbner basis with respect to the target ordering since truncated polynomials may miss some leading monomials.The new algorithm has been implemented in Maple and its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches in Maple. In practice, a Gröbner basis with respect to a target ordering can be computed in a single iteration on most examples.Since the proposed basis conversion algorithm uses simple concepts of Gröbner basis theory, it can be easily taught in contrast to methods based on Gröbner walk.

Graded Weakly Strongly Quasi-Primary Ideals over Commutative Graded Rings

In this article, we introduce and examine the concept of graded weakly strongly quasi primary ideals. A proper graded ideal P of R is said to be a graded weakly strongly quasi primary (shortly, Gwsq-primary) ideal if whenever 0≠xy∈P, for some homogeneous elements x,y∈R, then x2∈P or yn∈P, for some positive integer n. Many examples and properties of Gwsq-primary ideals are given. Among several results, we compare Gwsq-primary ideals and other classical graded ideals such as graded strongly quasi primary ideals, graded weakly primary ideals and graded weakly 2-prime ideals etc.

Homological Properties of Ideals Generated by Fold Products of Linear Forms

Homological Properties of Ideals Generated by Fold Products of Linear Forms

The sequentially Cohen–Macaulay property of edge ideals of edge-weighted graphs

The sequentially Cohen–Macaulay property of edge ideals of edge-weighted graphs

Action of higher derivations on semiprime rings

Abstract Let m , n {m,n} be the fixed positive integers and let ℛ {\mathcal{R}} be a ring. In 1978, Herstein proved that a 2-torsion free prime ring ℛ {\mathcal{R}} is commutative if there is a nonzero derivation d of R such that [ d ⁢ ( ϱ ) , d ⁢ ( ξ ) ] = 0 {[d(\varrho),d(\xi)]=0} for all ϱ , ξ ∈ R {\varrho,\xi\in R} . In this article, we study the above mentioned classical result for higher derivations and describe the structure of semiprime rings by using the invariance property of prime ideals under higher derivations. Precisely, apart from proving some other important results, we prove the following. Let ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} and ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} be two higher derivations of semiprime ring ℛ {\mathcal{R}} such that [ d n ⁢ ( ϱ ) , g m ⁢ ( ξ ) ] ∈ Z ⁢ ( ℛ ) {[d_{n}(\varrho),g_{m}(\xi)]\in Z(\mathcal{R})} for all ϱ , ξ ∈ ℐ {\varrho,\xi\in\mathcal{I}} , where ℐ {\mathcal{I}} is an ideal of ℛ {\mathcal{R}} . Then either ℛ {\mathcal{R}} is commutative or some linear combination of ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero or some linear combination of ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero. We enrich our results with examples that show the necessity of their assumptions. Finally, we conclude our paper with a direction for further research.

The weak Lefschetz property and mixed multiplicities of monomial ideals

The weak Lefschetz property and mixed multiplicities of monomial ideals

On 2-absorbing ideals of non-commutative semirings

Let [Formula: see text] be a non-commutative semiring with [Formula: see text]. We study the notions of 2-absorbing and strong 2-absorbing ideals of [Formula: see text] and we show that if the semiring is commutative, then these two notions are the same. Also, we give an example to show that in general, these two notions are different. Many properties of (strong) 2-absorbing ideals are proved as a generalization to the results for those over rings. For example, we show that a proper subtractive ideal [Formula: see text] of a semiring [Formula: see text] is a 2-absorbing ideal of [Formula: see text] if and only if whenever [Formula: see text], [Formula: see text] and [Formula: see text] are left ideals of [Formula: see text] such that [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text].

Newton iteration for lexicographic Gröbner bases in two variables

We present an m-adic Newton iteration with quadratic convergence for lexicographic Gröbner basis of zero dimensional ideals in two variables. We rely on a structural result about the syzygies in such a basis due to Conca and Valla, that allowed them to explicitly describe these Gröbner bases by affine parameters; our Newton iteration works directly with these parameters.

Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements

In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals of Levi graphs associated to some curve arrangements in the complex projective plane, like the d-arrangement of curves and the conic-line arrangements. We also discuss the existence of certain induced cycles in the Levi graphs of these arrangements and obtain lower bounds for the regularity of powers of the corresponding binomial edge ideals.

On classification of $7$-dimensional odd-nilpotent Leibniz algebras

In this paper we extend the method of canonical form for congruence of bilinear forms to give the classification of some subclasses of $7-$dimensional nilpotent Leibniz algebras. Odd-nilpotent Leibniz algebras are defined as that its even dimensional ideals in lower central series are all zero and the classification of $7-$dimensional complex odd-nilpotent Leibniz algebras with one dimensional Leib ideal is obtained by applying the aforementioned method.

Homological dimensions of Burch ideals, submodules and quotients

The notion of Burch ideals and Burch submodules were introduced (and studied) by Dao-Kobayashi-Takahashi in 2020 and Dey-Kobayashi in 2022 respectively. The aim of this article is to characterize various local rings in terms of homological invariants of Burch ideals, Burch submodules, or that of the corresponding quotients. Specific applications of our results include the following: Let (R,m) be a commutative Noetherian local ring. Let M=I be an integrally closed ideal of R such that depth(R/I)=0, or M=mN≠0 for some submodule N of a finitely generated R-module L such that either depth(N)⩾1 or L is free. It is shown that: (1) I has maximal projective (resp., injective) complexity and curvature. (2) R is Gorenstein if and only if ExtRn(M,R)=0 for any three consecutive values of n⩾max⁡{depth(R)−1,0}. (3) R is CM (Cohen-Macaulay) if and only if CM-dimR(M) is finite.

More Results on Intuitionistic Fuzzy Ideals of BE-algebras

This paper explores the intuitionistic fuzzy ideals in BE-algebras and establishes several new results related to their structure. We investigate the fundamental concepts and properties of intuitionistic fuzzy ideals and provide characterizations of an intuitionistic fuzzy ideal in BE-algebras. Our study focuses on examining the fundamental concepts and properties of these ideals and provides characterizations of intuitionistic fuzzy ideals in BE-algebras.

Multi-polar Q-hesitant Fuzzy Soft Implicative and Positive Implicative Ideals in BCK/BCI-algebras.

This paper focuses on exploring restricted mathematical concepts within the domain of BCK/BCI-algebras, specifically delving into the intricate realm of Multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. BCK and BCI-algebras are pivotal structures in mathematical logic and algebraic systems, finding widespread applications in fields like computer science and artificial intelligence. Our contribution lies in the introduction and thorough investigation of the innovative notions of multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals, uniquely tailored for BCK/BCI-algebras. These ideals exhibit exceptional flexibility in managing uncertain and hesitant information, serving as potent tools for modeling and solvingreal-world problems characterized by imprecise or incomplete data. This study rigorously defines and explores the foundational properties of multi-polar Q-hesitant fuzzy soft implicative ideals, underscoring their relevance and applicability within BCK/BCI-algebras. Additionally, we present the concept of positive implicative ideals, establishing their interconnectedness with multi-polar Q-hesitant fuzzy soft implicative ideals. Our investigation delves into these ideals’ algebraic and logical facets, offering valuable insights into their mutual interactions and significance within the context of BCK/BCI-algebras. To facilitate practical implementation, we develop algorithms and methodologies for identifying and characterizing multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. These computational tools enable efficient decision-making in scenariosinvolving uncertainty. Through illustrative examples and case studies, we showcase the potential of these ideals in handling complex, uncertain information, demonstrating their efficacy in aiding problem-solving processes. This research contributes significantly to advancing BCK/BCI-algebra theory by introducing innovative mathematical structures that bridge the gap between fuzzy logic, soft computing, and implicative ideals. The proposed multi-polar Q-hesitant fuzzy soft implicativeand positive implicative ideals open new avenues for addressing real-world problems characterized by imprecision and uncertainty. As such, they represent a valuable addition to the field of algebraic structures and their applications.

Dual ideal theory on L-algebras

<abstract><p>This paper aims to study bounded algebras in another perspective-dual ideals of bounded $ L $-algebras. As the dual concept of ideals in $ L $-algebras, dual ideals are designed to characterize some significant properties of bounded $ L $-algebras. We begin by providing a definition of dual ideals and discussing the relationships between ideals and dual ideals. Then, we prove that these dual ideals induce congruence relations and quotient $ L $-algebras on bounded $ L $-algebras. Naturally, in order to construct the first isomorphism theorem between bounded $ L $-algebras, the relationship between dual ideals and morphisms between bounded $ L $-algebras is investigated and that the kernels of any morphisms between bounded $ L $-algebras are dual ideals is proven. Fortunately, although the first isomorphism theorem between arbitrary bounded $ L $-algebras fails to be proven when using dual ideals, the theorem was proven when the range of morphism was good. Another main purpose of this study is to use dual ideals to characterize several kinds of bounded $ L $-algebras. Therefore, first, the properties of dual ideals in some special bounded $ L $-algebras are studied; then, some special bounded $ L $-algebras are characterized by dual ideals. For example, a good $ L $-algebra is a $ CL $-algebra if and only if every dual ideal is $ C $ dual ideal is proven.</p></abstract>

Some properties of neutrosophic nil radicals of neutrosophic ideals in rings

In this paper, we consider the notion of neutrosophic nil radicals of neutrosophic ideals in commutative rings and some properties of such nil radicals. Finally, we study the properties of semiprime neutrosophic ideals of rings.

Neutrosophic 𝔑-Structures in Semimodules over Semirings

The study of symmetry is a fascinating and unifying subject that connects various areas of mathematics in the twenty-first century. Algebraic structures offer a framework for comprehending the symmetries of geometric objects in pure mathematics. This paper introduces new concepts in algebraic structures, concentrating on semimodules over semirings and analysing the neutrosophic structure in this context. We explore the properties of neutrosophic subsemimodules and neutrosophic ideals after defining them. We discuss, utilizing neutrosophic products, the representations of neutrosophic ideals and subsemimodules, as well as the relationship between neutrosophic products and intersections. Finally, we derive equivalent criteria in terms of neutrosophic structures for a semiring to be fully idempotent.

Principals’ Views on the Factors Facilitating Idiosyncratic Deals They Make with Teachers

Non-standard, personalized arrangements between employees and their managers that are not granted to other subordinates are referred to in the literature as "idiosyncratic deals" (i-deals for short). Although the factors that influence the realization of these agreements have been researched in various sectors, it is apparent that these factors have not yet been uncovered in the school setting. Because of this deficiency in the literature, the purpose of this multiple case study was to identify the factors that facilitate making i-deals between principals and teachers based on the perceptions of sixteen principals working in public and private schools. Semi-structured interview questions were developed, and findings were reported based on the researcher-developed framework of six dimensions of (1) professional development i-deals, (2) task flexibility i-deals, (3) schedule flexibility i-deals, (4) location flexibility i-deals, (5) reduced workload i-deals, and (6) pay-related i-deals. Implications for research and application were also discussed based on the results.