Properties Of Ideals Research Articles

New results on Congruence Boolean Lifting Property

The Lifting Idempotent Property (LIP) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures (MV-algebras, commutative l-groups, BL-algebras, bounded distributive lattices, residuated lattices, etc.), as well as for some types of universal algebras (C. Muresan and the author defined and studied the Congruence Boolean Lifting Property (CBLP) for congruence modular algebras). A lifting ideal of a ring R is an ideal of R fulfilling LIP. In a recent paper, Tarizadeh and Sharma obtained new results on lifting ideals in commutative rings. The aim of this paper is to extend an important part of their results to congruences with CBLP in semidegenerate congruence modular algebras. The reticulation of such algebra will play an important role in our investigations (recall that the reticulation of a congruence modular algebra A is a bounded distributive lattice L(A) whose prime spectrum is homeomorphic with Agliano's prime spectrum of A). Almost all results regarding CBLP are obtained in the setting of semidegenerate congruence modular algebras having the property that the reticulations preserve the Boolean center. The paper contains several properties of congruences with CBLP. Among the results we mention a characterization theorem of congruences with CBLP. We achieve various conditions that ensure CBLP. Our results can be applied to a lot of types of concrete structures: commutative rings, l-groups, distributive lattices, MV-algebras, BL-algebras, residuated lattices, etc.

Neutrosophic κ-Structures in Ordered Semigroups

In general, ordered algebraic structures, particularly ordered semigroups, play an important role in fuzzification in many applied areas, such as computer science, formal languages, coding theory, error correction, etc. Nowadays, the concept of ambiguity is important in dealing with a variety of issues related to engineering modeling problems, network theory, decision-making problems in real-life situations, and so on. Several theories have been developed by various researchers to overcome the difficulties that arise from uncertainty, including fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and many more. In this paper, we focus solely on neutrosophic set theory. In ordered semigroups, we define and investigate the properties of neutrosophic κ-ideals and neutrosophic κ-interior ideals. We also use neutrosophic κ-ideals and neutrosophic κ-interior ideals to characterize ordered semigroups.

On some algebraic properties of edge ideals of ladder graphs

For any finite simple graph G one can associate a simplicial complex and an edge ideal I(G). We attempt to use the combinatorial properties of the given graph to understand some of the algebraic properties of the associated edge ideal. In particular, we study some important algebraic properties like graded Betti numbers, Castelnuovo–Mumford regularity and projective dimension of the edge ideal associated to the given ladder graph. We will introduce the new concept of A-type subgraphs and edge-disjoint A-type subgraphs to reduce the complexity of computing the regularity of a ladder graph. We are also able to find the projective dimension of a ladder graph using the Nguyen–Vu and Kimura’s works on the projective dimension of edge ideals.

Domains whose ideals meet a universal restriction

Let [Formula: see text] represent a set of proper nonzero ideals [Formula: see text] (respectively, t-ideals [Formula: see text]) of an integral domain [Formula: see text] and let P be a valid property of ideals of D. We say S(D) meets P (denoted [Formula: see text] if each [Formula: see text] is contained in an ideal satisfying P. If S(D) [Formula: see text] [Formula: see text] cannot be controlled. When [Formula: see text] [Formula: see text] [Formula: see text] does not imply I(R) [Formula: see text] while [Formula: see text] [Formula: see text] implies [Formula: see text] [Formula: see text] usually. We say S(D) meets P with a twist [Formula: see text]written [Formula: see text] if each [Formula: see text] is such that, for some [Formula: see text] [Formula: see text] is contained in an ideal satisfying P and study [Formula: see text] as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains.

N Sequence Prime Ideals

In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

Double centralisers and annihilator ideals of Young permutation modules

We study the double centraliser algebras and annihilator ideals of certain permutation modules for symmetric groups and their quantum analogues. Of particular interest is the behaviour of these algebras and ideals under base change. By exploiting connections with polynomial representation theory over a field we establish the double centraliser property and dimension invariance of annihilator ideals over fields. By elementary localisation techniques we obtain the corresponding results over arbitrary rings. The modules considered include the permutation modules relevant to partition algebras, as studied in the recent paper by Bowman, Doty and Martin, [3], and we obtain in particular their main results on the double centraliser property in this context. Much of our development is equally valid for the corresponding modules over Hecke algebras of type A and, for the most part, we give it at this level of generality.

Homogeneous polynomials and spurious local minima on the unit sphere

We consider forms on the Euclidean unit sphere. We obtain a simple and complete characterization of all points that satisfies the standard second-order necessary condition of optimality. It is stated solely in terms of the value of (i) f, (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. We also characterize a class of degree-d forms with no spurious local minima on \(\mathbb {S}^{n-1}\) by using a property of gradient ideals in algebraic geometry.

Characteristic-free test ideals

Tight closure test ideals have been central to the classification of singularities in rings of characteristic p > 0 p>0 , and via reduction to characteristic p > 0 p>0 , in equal characteristic 0 as well. Their properties and applications have been described by Schwede and Tucker [Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012]. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.

S-primary ideals of a commutative ring

Let R be a commutative ring with identity and a multiplicative subset. We define a proper ideal P of R disjoint from S to be S-primary if there exists an such that for all if then or We show that S-primary ideals enjoy analogs of many properties of primary ideals and we study the form of S-primary ideals of the amalgamation of A with B along an ideal J with respect to f (denoted by ), introduced and studied by D’Anna et al. S-primary ideals of the form of the trivial ring extensions and S-primary ideals of the form and of the bi-amalgamations are characterized.

Prime Radicals of Directed Pseudo-Ordered Rings

Characteristics of partially pseudo-ordered (K-ordered) rings are considered. Properties of ideals in pseudo-ordered rings are described. A variation of the concept of the prime radical for a ring in the subclass of directed pseudo-ordered rings (AO-ordered rings) is investigated. The description of elements of AO-prime radicals for AO-ordered rings is obtained.

Nobusawa gamma nearness hemirings

The aim of this paper is to introduce Nobusawa [Formula: see text]-nearness hemiring theory and also to study some properties of ideals and prime (semiprime) ideals for Nobusawa [Formula: see text]-nearness hemirings.

On $n$-Absorbing Ideals in a Lattice

Let L be a lattice, and let n be a positive integer. In this article, we introduce n-absorbing ideals in L. We give some properties of such ideals. We show that every n-absorbing ideal I of L has at most n minimal prime ideals. Also, we give some properties of 2-absorbing and weakly 2-absorbing ideals in L. In particular we show that in every non-zero distributive lattice L, 2-absorbing and weakly 2-absorbing ideals are equivalent.

Certain homological invariants of bipartite kneser graphs

In this paper, we obtain a combinatorial formula for computing the Betti numbers in the linear strand of edge ideals of bipartite Kneser graphs. We deduce lower and upper bounds for regularity of powers of edge ideals of these graphs in terms of associated combinatorial data and show that the lower bound is attained in some cases. Also, we obtain bounds on the projective dimension of edge ideals of these graphs in terms of combinatorial data.

Quasi ideals of nearness semirings

This article introduces quasi-ideals in semirings on weak nearness approximation spaces. Concepts and definitions are given to clarify the subject of quasi ideals in semirings on weak nearness approximation spaces. Some basic properties of quasi ideals are also given. Furthermore, it is given that the definition of upper-near quasi ideals. And, it is examined that the relationship between quasi ideals and upper near quasi ideals. Therefore, the features described in this study will contribute greatly to the theoretical development of the nearness semirings theory.

Symmetric bi-interior ideals of Symmetric Semigroups

In this paper, as a further generalization of ideals, we introduce the notion of symmetric bi-interior ideal as a generalization of symmetric quasi ideal, symmetric bi-ideal and symmetric interior ideal of symmetric semigroup and study the properties of symmetric bi-interior ideals of symmetric semigroup and simple symmetric semigroup.

Intuitionistic anti fuzzy normed ideals

In this paper, we introduce the notion of intuitionistic anti fuzzy normed subrings and intuitionistic anti fuzzy normed ideals and various properties of intuitionistic anti fuzzy normal subrings and ideals are examined. Also, we prove that the intersection of two intuitionistic anti fuzzy normed ideals is also an intuitionistic anti fuzzy normed ideal. Besides, we define the anti level subsets and identify the relationship between these sets and intuitionistic anti fuzzy normed ideals. In addition, we define the anti-image of intuitionistic anti fuzzy normed ideals and discuss the inverse image and anti-image for intuitionistic anti fuzzy normed ideals under epimorphism mapping.

Algebraic Properties of Edge Ideals of Some Vertex-Weighted Oriented Cyclic Graphs

We provide some exact formulas for the projective dimension and regularity of edge ideals associated to some vertex-weighted oriented cyclic graphs with a common vertex or edge. These formulas are functions in the weight of the vertices, and the numbers of edges and cycles. Some examples show that these formulas are related to direction selection and the assumption that [Formula: see text] for any vertex [Formula: see text] cannot be dropped.

Minimal prime ideals in seminearrings

This paper deals with the properties of ideals in seminearrings with special reference to right duo seminearrings. We formulate the concept of prime ideal and minimal prime ideal in a seminearring. In the case of Noetherian right duo seminearring, we derive some additional properties.

Ideals and Bosbach States on Residuated Lattices

In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

Algebraic Properties of Edge Ideals and Cover Ideals of Unbalanced Crown Graphs

In this paper, we introduce the unbalanced crown graphs $${\mathcal {C}}_{m,n}$$ , and compute all graded Betti numbers of edge ideals of these graphs in terms of associated numerical data. As a consequence, we determine the regularity and the projective dimension of edge ideals of these graphs. Also, the Hilbert series of the cover ideals of these graphs is obtained.