Related papers: On reduction number of products of ideals

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The aim of this paper is to extend the notion of quasi $J$-ideals of commutative rings to quasi $J$-submodules of modules. We call a proper submodule $N$ of $M$ a…

One proves that each almost local-global semihereditary ring has the stacked basis property and is almost Bezout. If M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annhilators is…

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal…

This work concerns the linearity defect of a module $M$ over a noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted by $\text{ld}_R M$. Roughly speaking, $\text{ld}_R M$ is the homological degree beyond which the…

Using commutative algebra methods we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a…

Let $A = \mathbb{F}_q[T]$, $\mathfrak{p} \subset A$ prime, $f(x) \in A[x]$ irreducible and set $R = A[x]/f(x)$. Denote its completion by $R_\mathfrak{p}$. The ideal class monoid $\text{ICM}(R_\mathfrak{p})$ is the set of fractional…

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to…

Set $ A := Q/({\bf z}) $, where $ Q $ is a polynomial ring over a field, and $ {\bf z} = z_1,\ldots,z_c $ is a homogeneous $ Q $-regular sequence. Let $ M $ and $ N $ be finitely generated graded $ A $-modules, and $ I $ be a homogeneous…

If O is a reduced operad in symmetric spectra, an O-algebra I can be viewed as analogous to the augmentation ideal of an augmented algebra. Implicit in the literature on Topological Andre-Quillen homology is that such an I admits a…

Let $S$ be a finitely generated standard multigraded algebra over an Artinian local ring $A$; $M$ a finitely generated multigraded $S$-module. This paper answers to the question when mixed multiplicities of $M$ are positive and…

For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if…

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed-Muller-type codes over finite fields. This gives…

Let $K$ be a field and let $S=\bigoplus_{n\geq 0} S_n$ be a positively graded $K$-algebra. Given $M=\bigoplus_{n\geq 0} M_n$, a finitely generated graded $S$-module, and $w>0$, we introduce the function $\zeta_M(z,w):=…

This paper deals with the study of the behaviour of the value semigroup of a curve singularity define over a global field reduced modulo a maximal ideal. We also define a global zeta function of the curve by means of motivic integration…

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0…

We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fr\'echet ideal. We prove that such $M$ is $2^{\aleph_0}$-saturated if its theory is stable and not…

Let $k$ be a field of characteristic zero, and $R=k[x_1, \ldots, x_d]$ with $d \geq 3$ be a polynomial ring in $d$ variables. Let $\m=(x_1, \ldots, x_d)$ be the homogeneous maximal ideal of $R$. Let $\mathcal{K}$ be the kernel of the…

Let $R=\bigoplus_{n\in \NN_0}R_n$ be a standard graded ring, $R_+=\bigoplus_{n\in \NN}R_n$ its irrelevant ideal, and $M$ a finitely generated graded $R$-module. In this paper, we study the asymptotic behavior of the sequence $\{\mu^i(\p_0,…

Let $K$ be a field and let $R = K[X_1, \ldots, X_m]$ with $m \geq 2$. Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1 \leq g \leq m -1$. Suppose $H^i_I(R) \neq 0$ for some $i \geq 0$. We show (1)…

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R is always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over…