scholarly journals Monomial Ideals with Primary Components given by Powers of Monomial Prime Ideals

10.37236/3813 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Jürgen Herzog ◽  
Marius Vladoiu
Keyword(s):  
Monomial Ideals ◽  
Prime Ideals ◽  
Polymatroidal Ideals

We characterize monomial ideals which are intersections of powers of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.

scholarly journals The stable property of projective dimension

2019 ◽  
Vol 18 (12) ◽  
pp. 1950224
Author(s):  
Somayeh Bandari ◽  
Raheleh Jafari
Keyword(s):  
Projective Dimension ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Stable Property ◽  
Polymatroidal Ideals

We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen–Macaulay property. Indeed, we study the class of monomial ideals [Formula: see text], whose projective dimension is stable under monomial localizations at monomial prime ideals [Formula: see text], with [Formula: see text]. We study the relations between this property and other sorts of Cohen–Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.

scholarly journals A family of monomial ideals with the persistence property

2019 ◽  
Vol 18 (05) ◽  
pp. 1950093
Author(s):  
Somayeh Moradi ◽  
Masoomeh Rahimbeigi ◽  
Fahimeh Khosh-Ahang ◽  
Ali Soleyman Jahan
Keyword(s):  
Monomial Ideal ◽  
Independent Set ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Stable Set ◽  
Ideal Formula ◽  
Path Graph ◽  
Associated Prime Ideals ◽  
Persistence Property ◽  
Positive Integers

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers [Formula: see text] and [Formula: see text], we consider the monomial ideal [Formula: see text] generated by all monomials [Formula: see text], where [Formula: see text] is an independent set of vertices of the path graph [Formula: see text] of size [Formula: see text], which is indeed the facet ideal of the [Formula: see text]th skeleton of the independence complex of [Formula: see text]. We describe the set of associated primes of all powers of [Formula: see text] explicitly. It turns out that any such ideal [Formula: see text] has the persistence property. Moreover, the index of stability of [Formula: see text] and the stable set of associated prime ideals of [Formula: see text] are determined.

Persistence property for some classes of monomial ideals of a polynomial ring

2017 ◽  
Vol 16 (06) ◽  
pp. 1750105 ◽  
Author(s):  
Mehrdad Nasernejad
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Associated Prime Ideals ◽  
Persistence Property ◽  
Associated Prime

Let [Formula: see text] be a field and [Formula: see text] be a polynomial ring in the variables [Formula: see text]. In this paper, we introduce two classes of monomial ideals of [Formula: see text], which have the following properties: (i) The (strong) persistence property of associated prime ideals. (ii) There exists a strongly superficial element. (iii) Ratliff–Rush closed. Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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