Related papers: Monomial ideals with primary components given by p…
We give explicit formulas for the Bhattacharya function of m-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals.
Let R be a commutative ring with1 and R[X] be the polynomial ring over R. We determine the underlying ring R over which the polynomial ring R[X] has the property that all its prime ideals are set theoretic complete intersections.
Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. Monoid congruences (and…
In our earlier article~\cite{CanSakran} we initiated a study of the complement-finite submonoids of the group of integer points of a unipotent linear algebraic group. In the present article, we continue to develop tools and techniques for…
The principal result is a primary decomposition of ideals generated by the (2x2)-subpermanents of a generic matrix. These permanental ideals almost always have embedded components and their minimal primes are of three distinct heights. Thus…
This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative…
We establish characteristic-free criteria for the componentwise linearity of graded ideals. As applications, we classify the componentwise linear ideals among the Gorenstein ideals, the standard determinantal ideals, and the ideals…
We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1].…
To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…
Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions derived from simplicial complexes. For a generic monomial ideal, the associated primes satisfy a saturated…
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the…
The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see,…
The support poset of a monomial ideal $I\subseteq\mathbf{k}[x_1,\dots,x_n]$ encodes the relation between the variables $x_1,\dots,x_n$ and the minimal monomial generators of $I$. It is known that not every poset is realizable as the support…
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…
We characterize the monomial ideals $I\subset K[x_1,\ldots,x_n]$ with the property that the polarization $I^p$ and $I^{\sigma^n}:=$ the ideal obtained from $I$ by the $n$-th iterated squarefree operator $\sigma$ are isomorphic via a…
We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to…
We study zero divisors and minimal prime ideals in semirings of characteristic one. Thereafter we find a counterexample to the most obvious version of primary decomposition, but are able to establish a weaker version. Lastly, we study…
Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field, generated by square free monomials. We show that some inequalities among the numbers of square free monomials of $I\setminus J$ of different degrees give upper bounds…
In this paper we introduce a family of monomial ideals with the persistence property. Given positive integers $n$ and $t$, we consider the monomial ideal $I=Ind_t(P_n)$ generated by all monomials $\textbf{x} ^F$, where $F$ is an independent…
We give a new combinatorial characterization of the big height of a squarefree monomial ideal leading to a new bound for the projective dimension of a monomial ideal.