Related papers: Regular sequences and ZD-modules

Let R be a commutative Noetherian d-dimensional complete equicharacterisitc regular local ring and let I be an ideal of R such that every minimal prime over I has height at most c. Let v=d - [(d-2)/c]-1 and v'=d - [(d-1)/c]. It has been…

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…

We consider modules $M$ over Lie algebroids ${\mathfrak g}_A$ which are of finite type over a local noetherian ring $A$. Using ideals $J\subset A$ such that ${\mathfrak g}_A \cdot J\subset J $ and the length $\ell_{{\mathfrak g}_A}(M/JM)<…

Let $M$ be an $R$-module over a Noetherian ring $R$ and $\mathfrak{a}$ be an ideal of $R$ with $c={\rm cd}(\mathfrak{a},M)$. First, we prove that $M$ is finite $\mathfrak{a}$-relative Cohen-Macaulay if and only if ${\rm…

Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $I$ a proper ideal of $R$. In this paper we introduce and analyze some properties of $r(I, M)=\bigcup_{k\geqslant 1} (I^{k+1}M: I^kM)$, {\it the Ratliff-Rush…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M, N$ two finitely generated $R$-modules. By using a spectral sequence argument, it is shown that if either $\mathrm{dim}_RM\leq2$ and $\mathrm{H}^{i}_\mathfrak{a}(N)$…

In this paper, we introduce and study the class $S$-$\mathcal{F}$-ML of $S$-Mittag-Leffler modules with respect to all flat modules. We show that a ring $R$ is $S$-coherent if and only if $S$-$\mathcal{F}$-ML is closed under submodules. As…

Let $\mathfrak{q}$ denote an ideal in a Noetherian local ring $(A,\mathfrak{m})$. Let $\underline{a}=a_1,\ldots,a_d \subset \mathfrak{q}$ denote a system of parameters in a finitely generated $A$-module $M$. This note investigate an…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $X$ a finite $R$-module. We prove that the category of $\fa$-cominimax modules is a Melkersson subcategory of $R$-modules whenever $\dim R\leq…

A depiction of a nonnoetherian integral domain $R$ is a special coordinate ring that provides a framework for describing the geometry of $R$. We show that if $R$ is noetherian in codimension 1, then $R$ has a unique maximal depiction $T$.…

Let $(R, \mathfrak{m})$ be a $d$-dimensional Noetherian local ring that is formally equidimensional, and let $M$ be an arbitrary $R$-submodule of the free module $F = R^p$ with an analytic spread $s:=s(M)$. In this work, inspired by…

We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $i\gg 0$ implies that…

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}$ is bounded below by…

Let $R$ be a Cohen-Macaulay local ring and let $M$ and $N$ be non-zero finitely generated $R$-modules. We investigate necessary conditions for the depth formula $\depth(M)+\depth(N)=\depth(R)+\depth(M\otimes_{R}N)$ to hold. We show that,…

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…

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\mathrm{deg}\ x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form…

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form…

We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of…

We consider local non-Gorenstein rings of the form $(S_i,\mathfrak{n}_i)=k[X, Y_1, \ldots ,Y_i]/\left(X^2, (Y_1, \ldots, Y_i)^2\right), $ where $i\geq 2.$ We show that every totally reflexive $S_i$-module has a presentation matrix of the…

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…