Related papers: A note on the $v$-invariant

Let $I$ be a monomial ideal of a polynomial ring $R$. In this paper we determine a number $B$ such that $\Ass (I^n/I^{n+1}) = \Ass (I^{B}/I^{B+1})$ for all $n\geq B$.

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest…

Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. In this article, we prove that $$\mathrm{Ass}_R(M/I^{n} M) = \mathrm{Ass}_R(0:_{M} I) \cup \mathrm{Ass}_R(I^{n-1} M/I^{n} M) \text{ for all } n \gg…

Let $B$ be a Noetherian normal local ring, and $G\subset\Aut(B)$ a cyclic group of local automorphisms of prime order. Let $A$ be the ring of $G$-invariants of $B$, assume that $A$ is Noetherian. We study the invariant morphism; in…

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest…

Let $I$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$. The asymptotic behaviour of the $\text{v}$-number of the powers of $I$ is investigated. Natural lower and upper bounds which are linear…

The notion of Vasconcelos invariant, known in the literature as v-number, of a homogeneous ideal in a polynomial ring over a field was introduced in 2020 to study the asymptotic behaviour of the minimum distance of projective Reed-Muller…

Let $R$ be a commutative Noetherian $\mathbb{N}$-graded ring. Let $N\subseteq M$ be finitely generated $\mathbb{Z}$-graded $R$-modules. Let $I_1,\ldots,I_r$ be non-zero proper homogeneous ideals of $R$. Denote ${\bf…

Let $k$ be a field and $G \subseteq Gl_n(k)$ be a finite group with $|G|^{-1} \in k$. Let $G$ act linearly on $A = k[X_1, \ldots, X_n]$ and let $A^G$ be the ring of invariant's. Suppose there does not exist any non-trivial one-dimensional…

For an ideal $I$ in a Noetherian ring $R$, we introduce and study its conductor as a tool to explore the Rees algebra of $I$. The conductor of $I$ is an ideal $C(I)\subset R$ obtained from the defining ideals of the Rees algebra and the…

Let $R$ be a commutative ring with identity. In this note, we study the property: If $ I \subsetneqq J$ are ideals in $R$, then $ I^n \subsetneqq J^n$ for all $ n\geq 1$. We define the notion of a big ideal (Definition 1.2). It is noted…

Let $R$ denote a commutative Noetherian ring, $I$ an ideal of $R$, and let $S$ be a multiplicatively closed subset of $R$. In \cite{Ra1}, Ratliff showed that the sequence of sets ${\rm Ass}_RR/\bar{I}\subseteq {\rm Ass}_RR/\bar{I^2}…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length…

Let $A$ be a Noetherian ring, $J\subseteq A$ an ideal and $C$ a finitely generated $A$-module. In this note we would like to prove the following statement. Let $\{I_n\}_{n\geq 0}$ be a collection of ideals satisfying : (i) $I_n\supseteq…

In this article, we generalize the well-known result that ideals of Noetherian polynomial rings have only finitely many initial ideals to the situation of ascending ideal chains in non-Noetherian polynomial rings. More precisely, we study…

Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $Ass^i E = \{ Q \in \ Ass E \mid \ height Q = i \}$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R/P$…

Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal $I$ in a polynomial ring $S$, $\mathrm{v}(I^k)$ is a linear function in $k$ for $k>>0$. Later, Ficarra conjectured…

Let $R$ be a Noetherian ring, $I_1,\ldots,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S = \bigoplus_{\underline{n} \in \mathbb{N}^r} S_{\underline{n}}$ be a Noetherian standard $\mathbb{N}^r$-graded…