Related papers: The Goto numbers of parameter ideals

For a group $G$, N-series $\cal G$ of $G$ and commutative ring $R$ let $I^n_{R,\cal G}(G)$, $n\ge 0$, denote the filtration of the group algebra $R(G)$ induced by $\cal G$, and $I_R(G)$ its augmentation ideal. For subgroups $H$ of $G$, left…

Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality $\chi_1(Q;M)=\operatorname{hdeg}_Q(M)-\mathrm{e}_Q^0(M)$, where $\chi_1(Q;M)$,…

Let $R=k[|t^a,t^b,t^c|]$ be a complete intersection numerical semigroup ring over an infinite field $k$, where $a,b,c\in\BN$. The generalized Loewy length, which is Auslander's index in this case, is computed in terms of the minimal…

Perfect colourings of the rings of cyclotomic integers with class number one are studied. It is shown that all colourings induced by ideals (q) are chirally perfect, and vice versa. A necessary and sufficient condition for a colouring to be…

The covering number of a group $G$, denoted by $\sigma(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$…

For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We…

Let $R$ be a Cohen-Macaulay local ring possessing a canonical module. We compare the initial and terminal Betti numbers of modules in a series of nontrivial cases. We pay special attention to the Betti numbers of the canonical module. Also,…

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a…

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and…

This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed ideals. We also explore properties of such rings related to…

For a Noetherian local ring $(\RR, \m)$, the first two Hilbert coefficients, $e_0$ and $e_1$, of the $I$-adic filtration of an $\m$-primary ideal $I$ are known to code for properties of $\RR$, of the blowup of $\spec(\RR)$ along $V(I)$, and…

Let $(A,\mathfrak{m},\Bbbk)$ denote a local Noetherian ring and $\mathfrak{q}$ an ideal such that $\ell_A(M/\mathfrak{q}M) < \infty$ for a finitely generated $A$-module $M$. Let $\au = a_1,\ldots,a_d$ denote a system of parameters of $M$…

This paper presents algorithms for calculating the quadratic character and the norms of prime ideals in the ring of integers of any quadratic field. The norms of prime ideals are obtained by means of a sieve algorithm using the quadratic…

Let R be a two-dimensional regular local ring with maximal ideal \mathfrak m, and let \wp be a simple complete \mathfrak m-primary ideal which is residually rational. Let R_0:= R\subsetneqq ...\subsetneqq R_r be the quadratic sequence…

Let $R$ be a Noetherian local ring and $m$ a positive integer. Let $I$ be the ideal of $R$ generated by the maximal minors of an $m \times (m + 1)$ matrix $M$ with entries in $R$. Assuming that the grade of the ideal generated by the…

We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some…

This work concerns commutative algebras of the form $R=Q/I$, where $Q$ is a standard graded polynomial ring and $I$ is a homogenous ideal in $Q$. It has been proposed that when $R$ is Koszul the $i$th Betti number of $R$ over $Q$ is at most…

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,m) be a complete local ring, a an ideal of R and M a finitely generated R-module. The aim of this paper is to show that for any non-negative integer n, the least integer i such that the i-th local cohomology with respect to a is not…