Properties Of Ideals Research Articles

Cominimaxness with respect to ideals of dimension two and local cohomology

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] an [Formula: see text]-module with [Formula: see text]. We get equivalent conditions for top local cohomology module [Formula: see text] to be Artinian and [Formula: see text]-cofinite Artinian separately. In addition, we prove that if [Formula: see text] is a local ring such that [Formula: see text] is minimax, for each [Formula: see text], then [Formula: see text] is minimax [Formula: see text]-module for each [Formula: see text] and for each finitely generated [Formula: see text]-module [Formula: see text] with [Formula: see text] and [Formula: see text]. As a consequence we prove that if [Formula: see text] and [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax if (and only if) [Formula: see text], [Formula: see text] and [Formula: see text] are minimax. We also prove that if [Formula: see text] and [Formula: see text] such that [Formula: see text] is minimax for all [Formula: see text], then [Formula: see text] is [Formula: see text]-cominimax for all [Formula: see text] if (and only if) [Formula: see text] is minimax for all [Formula: see text].

Further Results on the Sharp Ordering in Rings

In this article, we will present some results relating sharp ordering and investigate the properties of one-sided cyclic ideals in rings. Necessary and sufficient conditions for the invertibility and the group invertibility of the linear combination are given, where a is below b under the sharp order, a and b are elements of an algebra, and c1, c2 are elements of an arbitrary field.

Nilpotent Ideals in Generalized Amenable Banach Algebras

In this article, some properties such as the approximation property of non-zero nilpotent ideals in generalized amenable Banach algebras are investigated. Also, we show that in pseudo-contractible Banach algebras, non-zero nilpotent ideals cannot have the approximation property.

On properties of hesitant fuzzy ideals in semigroups

In this paper, we study some properties of hesitant fuzzy ideals and hesitant fuzzy bi-ideals in a semigroup and discuss their characterizations. Also we introduce hesitant fuzzy interior ideals in a semigroup and studied their properties. It is proved that in a semigroup a hesitant fuzzy ideal is a hesitant fuzzy interior ideal but the converse is not true. Moreover we prove that in regular and in intra-regular semigroups the hesitant fuzzy ideals and the hesitant fuzzy interior ideals coincide.

A Note on Ideals and Regularity in Ternary Semigroups

In this paper we have studied some properties of ideals in ternary semigroups.We have also studied some properties of regular ternary semigroups.We have presented some definitions and prepositions related to them.

Codimension and projective dimension up to symmetry

AbstractSymmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non‐trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen–Macaulayness obstructions.

Lefschetz properties of monomial algebras with almost linear resolution

We study the WLP and SLP of artinian monomial ideals in via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of S/I is linear for at least n – 2 steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial algebras with almost linear resolution.

S-prime ideals of a commutative ring

Let R be a commutative ring with identity and $$S\subseteq R$$ a multiplicative subset. In this paper we introduce the concept of S-prime ideal which is a generalization of prime ideal. Let P be an ideal of R disjoint with S. We say that P is an S-prime ideal of R if there exists an $$s\in S$$ such that for all $$a,b\in R$$ if $$ab\in P,$$ then $$sa\in P$$ or $$sb\in P$$ . We show that S-prime ideals enjoy analogs of many properties of prime ideals and we study them over S-Noetherian rings.

Unmixedness and arithmetic properties of matroidal ideals

Let R=k[x_1,ldots ,x_n] be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that ara(I)=n-d+1. This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture).

Unexpected curves and Togliatti‐type surfaces

AbstractThe purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface of degree 7 in with its osculating spaces of order 2 which in every point of have dimension lower than expected. We put this result in perspective with earlier examples of surfaces with defective osculating spaces due to Shifrin and Togliatti. Our considerations are rendered by an analysis of Lefschetz Properties of ideals associated with the studied surfaces.

In the shadows of a hypergraph: looking for associated primes of powers of square-free monomial ideals

The aim of this paper is to study the associated primes of powers of square-free monomial ideals. Each square-free monomial ideal corresponds uniquely to a finite simple hypergraph via the cover ideal construction, and vice versa. Let H be a finite simple hypergraph and J(H) the cover ideal of H. We define the shadows of hypergraph, H, described as a collection of smaller hypergraphs related to H under some conditions. We then investigate how the shadows of H preserve information about the associated primes of the powers of J(H). Finally, we apply our findings on shadows to study the persistence property of square-free monomial ideals and construct some examples exhibiting failure of containment.

L-Fuzzy Ideals in Couple Γ-Semirings

Let M be a Γ-semiring. In this paper we obtain some properties of L-fuzzy ideals in M x M. Our results take inspiration from [M. Murali Krishna Rao and B. Vekateswarlu, L-fuzzy ideals in Γ-semiring, Annals of Fuzzy Mathematics and Informatics 10(1) (2015), 1-16]. The readers are left with a conjecture.

Bar Code and Janet-like division

Bar Codes are combinatorial objects encoding many properties of monomial ideals. In this paper we employ these objects to study Janet-like divisions. Given a finite set of terms U, from its Bar Code we can compute the Janet-like nonmultiplicative power of its elements and detect completeness of the set. Some observation on the computation of Janet-like bases conclude the work.

Interval-Valued Doubt Fuzzy Ideals in BCK-Algebras

Nearly forty years ago, interval-valued fuzzy sets were propounded by Zadeh as the normal ramification of fuzzy sets. This article focuses on the basics of a theory for such an interval-valued fuzzy set becoming interval-valued doubt fuzzy subalgebra and an interval-valued doubt fuzzy ideal of BCK-algebras. Also, the authors discuss fuzzy translation, fuzzy multiplication of an interval-valued doubt fuzzy subalgebra/ideal of a BCK-algebra. Besides this, the authors have attempted to substantiate a few common features relating them. At the same time, some properties of interval-valued doubt fuzzy ideals under homomorphism are investigated and the product of interval-valued doubt fuzzy ideals in BCK-algebras is also established.

Nil clean ideal

In this article, we introduce the notion of nil clean ideals of a ring. We define an ideal I of a ring R to be nil clean ideal if every element of I can be written as a sum of an idempotent and a nilpotent element of R. We discuss many interesting properties of nil clean ideals.

New Properties of Anti Fuzzy Ideals of Regular Semigroups

In this article, we study some properties of anti-fuzzy sub-semigroup, anti fuzzy left (right, two sided) ideal, anti fuzzy ideal, anti fuzzy generalized bi-ideal, anti fuzzy interior ideals and anti fuzzy two sided ideal of regular semigroup. Also, we characterized regular LA-semigroup in terms of their anti fuzzy ideal.

Asymptotic behavior of coefficient ideals

In this paper we study properties of the coefficient ideals of the powers of an arbitrary ideal in a quasi-unmixed local ring.

Lyubeznik tables of linked ideals

In this paper, we examine the Lyubeznik tables of two linked ideals [Formula: see text] and [Formula: see text] of a complete regular local ring [Formula: see text] containing a field. More precisely, we prove that the Lyubeznik tables of two evenly linked ideals [Formula: see text] and [Formula: see text] are the same when [Formula: see text] and [Formula: see text] both satisfy one of the following properties: (1) canonically Cohen–Macaulay, (2) generalized Cohen–Macaulay and (3) Buchsbaum. Furthermore, we give some conditions for equality of Lyubeznik tables of two linked ideals of dimension 2.

Projective dimension and regularity of edge ideals of some weighted oriented graphs

We provide formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. We derive from that exact formulas for the depth of those ideals. We also give some examples to show that the assumptions cannot be dropped.

Krull dimension and regularity of binomial edge ideals of block graphs

We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present linear time algorithms to compute the Castelnuovo–Mumford regularity and the Krull dimension of binomial edge ideals of block graphs.