Parametric Decomposition of Monomial Ideals, II

Communications in Algebra, 2013

Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[R d 0 ] where A is an arbitrary commutative ring with identity. We classify the irreducible elements of this set, which we call m-irreducible, and we classify the elements that admit decompositions into finite intersections of m-irreducible ideals.

In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.

Communications in Algebra, 1993

Bulletin of the Australian Mathematical Society, 2013

Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i...

Communications in Algebra, 2018

Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated radical ideals. In particular, we present some applications to the symbolic powers of square-free monomial ideals.

Journal of Pure and Applied Algebra, 2020

A commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains, noting the almost local nature of the property and, in the case of a local domain, relating it to the structure of its integral closure. In a 2011 paper, Ay and Klingler obtain similar results for Noetherian reduced rings. In this paper, we examine the UDI property for arbitrary commutative Noetherian rings, establishing the same almost local nature of the property, and giving an example which shows that the local results do not extend to commutative Noetherian rings in general.

Communications in Algebra, 2012

Let (S, n) be a Noetherian local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Assume that the associated graded ring gr n (S) of S with respect to n is a UFD. We examine generators of the leading form ideal I * of I in gr n (S) and prove that I * is a perfect ideal of gr n (S), if I * is 3-generated. Thus, in this case, letting R = S/I and m = n /I, if gr n (S) is Cohen-Macaulay, then gr m (R) = gr n (S)/I * is Cohen-Macaulay. As an application, we prove that if (R, m) is a one-dimensional Gorenstein local ring of embedding dimension 3, then gr m (R) is Cohen-Macaulay if the reduction number of m is at most 4.

Factorization in integral domains, 2017

Let x 1 ,. .. , x d be indeterminates over an infinite field F , let R denote the polynomial ring F [x 1 ,. .. , x d ], and let M denote the maximal ideal (x 1 ,. .. , x d)R. If I is an M-primary ideal the Hilbert polynomial P I (n) = e 0 (I) n + d − 1 d − e 1 (I) n + d − 2 d − 1 + • • • + (−1) d e d (I) gives the length of the R-module R/I n for sufficiently large positive integers n. The integral closure I of I is the unique largest ideal of R containing I and having the same coefficient e 0 (i.e., multiplicity) as I, and the Ratliff-Rush ideal I of I is the unique largest ideal containing I and having the same Hilbert polynomial as I. Kishor Shah has shown in [S1] that there exists a unique chain of ideals 1 I ⊆ I = I {d} ⊆ • • • ⊆ I {k} ⊆ • • • ⊆ I {0} = I , where, for 0 ≤ k ≤ d, the ideal I {k} is maximal with the property of having the same coefficients e 0 ,. .. , e k of its Hilbert polynomial as those of I. The ideal I {k} is called the k-th coefficient ideal of I. If I = I {k} , we say I is an e k-ideal. We are particularly interested in the case where R is of dimension two. In this setting, an M-primary ideal I has reduction number at most one (i.e., if J is a minimal reduction of I, then JI = I 2) if and only if the Rees algebra R[It] is Cohen-Macaulay [HM, Prop. 2.6],[JV, Theorem 4.1], or [S2, Corollary 4(f)]. Moreover, the coefficients e 1 (I) and e 2 (I) are nonnegative, and it follows from [Hu, Theorem 2.1] that I has reduction number at most one if and only if λ(R/I) = e 0 (I) − e 1 (I), and if this holds, then e 2 (I) = 0. We say that an ideal with these 1 The existence of this unique chain of ideals is shown in [S1, Theorem 1] for an ideal primary for the maximal ideal of a quasi-unmixed local ring with infinite residue field. Since R M is regular and so, in particular, quasi-unmixed, and since the length of R/I n is equal to the length of R M /I n R M , Shah's result also applies in our setting.

Archiv der Mathematik

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound µ(I 2 ) ≥ 9 for the number of minimal generators of I 2 . Recently, Gasanova constructed monomial ideals such that µ(I) > µ(I n ) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that µ(I) > µ(I 2 ) > • • • > µ(I n ) = (n + 1) 2 for any positive integer n, which provides one of the most unexpected behaviors of the function µ(I k ).

In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients.