Related papers: On reduction number of products of ideals

Recently, new classes of positive and measurable functions, $\mathcal{M}(\rho)$ and $\mathcal{M}(\pm \infty)$, have been defined in terms of their asymptotic behaviour at infinity, when normalized by a logarithm (Cadena et al., 2015, 2016,…

Let $\frak g$ be a semisimple Lie algebra and $\frak k\subset\frak g$ be a reductive subalgebra. We say that a $\frak g$-module $M$ is a bounded $(\frak g, \frak k)$-module if $M$ is a direct sum of simple finite-dimensional $\frak…

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely generated graded $S$-module, the Hilbert coefficients $e_i(M/I^kM)$ are polynomial…

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R)^*$ be the set of all non-trivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I(R)^*$, and two…

We study reducing invariants of modules related to certain homological properties. For modules of finite reducing projective dimension, we establish grade inequalities. We prove that if $\mathbb{P}$ is the (uniform) Auslander condition, or…

Let $A$ be a noetherian ring, $I$ an ideal of $A$ and $N\subset M$ finitely generated $A$-modules. The relation type of $I$ with respect to $M$, denoted by ${\bf rt}\,(I;M)$, is the maximal degree in a minimal generating set of relations of…

Let $C \subset {\bf N}^d$ be an affine semigroup, and $R=K[C]$ its semigroup ring. This paper is a collection of various results on "$C$-graded" $R$-modules, especially, monomial ideals. For example, we show the following: If $R$ is normal…

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give…

The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C^*$-modules and their maps. Here we give a…

For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with…

Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-primary ideal, $I_1,...,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this…

Let $R$ be a commutative ring with identity and $M$ a unitary $R$-module. The purpose of this paper is to introduce the concept of semi-$n$-submodules as an extension of semi $n$-ideals and $n$-submodules. A proper submodule $N$ of $M$ is…

The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some…

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is…

For the algebra $I_n$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $I_n$-modules is given. It is proven that the category of…

Given a finitely generated multiplicative subgroup of rational numbers $\Gamma$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group…

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. Let $\mathcal{P}$ be the class of all $I$-generated $R$-modules $M$ (i.e. there is an epimorphism $I^{(\Lambda)} \twoheadrightarrow M$) and let…

We introduce and study the arithmetic function E_m(n), defined as the sum of the remainders of n when divided by the first m positive integers. Although the definition is elementary, the function encodes rich arithmetic structure. In this…

Let $\mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $\mathrm{Supp}_RM\subseteq\mathrm{V}(\mathfrak{a})$. We show that if $\mathrm{dim}_RM\leq2$, then $M$ is $\mathfrak{a}$-cofinite if…

Let $A$ be a Noetherian standard $\mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,\ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $\mathbb{N}$-graded $A$-module. We prove that there exist two…