Related papers: Quasi-socle ideals in a Gorenstein local ring

We study reflexive ideals in one-dimensional Cohen-Macaulay local rings, providing characterizations of almost Gorenstein rings, rings with minimal multiplicity, and Arf rings, which describe their reflexive fractional ideals.

Let $\bar{I}$ denote the integral closure of an ideal in a Noetherian ring $R$. The main result of this paper asserts that $R$ is locally quasi-unmixed if and only if, the topologies defined by $\overline{I^n}$ and $I^{\langle n\rangle}$,…

In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems…

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$ with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For $0 \leq i \leq d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let…

Let Q be a parameter ideal of a Noetherian local ring (R,m). The Goto number g(Q) of Q is the largest integer g such that Q:m^g is integral over Q. We examine the values of g(Q) as Q varies over the parameter ideals of R. We concentrate…

We investigate the nearly Gorenstein property of a local ring defined by the maximal minors of a specific $2 \times n$ matrix with entries in the formal power series ring $k[[X_1, X_2, \ldots , X_n]]$ over a field $k$. Our findings allow us…

In two dimensional regular local rings integrally closed ideals have a unique factorization property and have a Cohen-Macaulay associated graded ring. In higher dimension these properties do not hold for general integrally closed ideals and…

In this work we describe the local cohomology of reflexive modules of rank one over normal semigroup rings with respect to monomial ideals. Using our description we show that the problem of classifying maximal Cohen-Macaulay modules of rank…

Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every…

In this paper we study Ulrich ideals of and Ulrich modules over Cohen--Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and…

A commutative noetherian local ring $(R,\mathfrak{m})$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there…

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study the depths of $F(I)$ under certain depth assumptions on $G(I)$ and length condition on quotients of…

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $\omega_R$ and the property that the socle…

We provide a characterization of one-dimensional almost Gorenstein rings in terms of the trace ideal. As an application, we investigate the almost Gorenstein property of certain $\mathbb{Z}_2$-graded rings.

Let $R$ be a complete local Gorenstein ring of dimension one, with maximal ideal m. We show that if $M$ is a Cohen-Macaulay $R$-module which begins an AR-sequence, then this sequence is produced by a particular endomorphism of m…

Let R be a Cohen-Macaulay local ring possessing a canonical module. In this paper we consider when the maximal ideal of R is self-dual, i.e. it is isomorphic to its canonical dual as an R-module. local rings satisfying this condition are…

We studies the nearly Gorenstein property for Veronese subalgebras of (semi-)standard graded algebras. We introduce a condition~$(\natural)$ for Cohen--Macaulay semi-standard graded rings, motivated by the study of Ehrhart rings. We show…

The notion of 2-AGL ring in dimension one which is a natural generalization of almost Gorenstein local ring is posed in terms of the rank of Sally modules of canonical ideals. The basic theory is developed, investigating also the case where…

The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times_T S$ of Cohen-Macaulay local…

The purpose of this paper is to investigate a relationship between the index of reducibility and the Chern coefficient for primary ideals. Therefore, the main result of this paper gives a characterization of a Cohen-Macaulay ring in terms…