Related papers: How many ideals whose quotient rings are Gorenstei…
Given a Noetherian local ring (R,m) it is shown that there exists an integer l such that R is Gorenstein if and only if some system of parameters contained in m^l generates an irreducible ideal. We obtain as a corollary that R is Gorenstein…
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…
We give a criterion for almost Gorenstein property for semigroup rings associated with simplicial semigroups. We extend Nari's theorem for almost symmetric numerical semigroups to simplicial semigroups with higher rank. By this criterion,…
Starting with a commutative ring $R$ and an ideal $I$, it is possible to define a family of rings $R(I)_{a,b}$, with $a,b \in R$, as quotients of the Rees algebra $\oplus_{n \geq 0} I^nt^n$; among the rings appearing in this family we find…
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.
We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the…
In 1983 Kustin and Miller introduced a construction of Gorenstein ideals in local Gorenstein rings, starting from smaller such ideals. We review and modify their construction in the case of graded rings and discuss it within the framework…
Let $H\subseteq \mathbb{N}^d$ be a normal affine semigroup, $R=K[H]$ its semigroup ring over the field $K$ and $\omega_R$ its canonical module. The Ulrich elements for $H$ are those $h$ in $H$ such that for the multiplication map by…
In this paper, we view the collection of ideals of a commutative principal ideal ring from two perspectives: one as an ordered semigroup I(R) and the other as a category I_R . It is shown that I(R) is a regular ordered semigroup whereas I_R…
As part of stratification of Cohen-Macaulay rings, we introduce and develop the theory of Goto rings, generalizing the notion of almost Gorenstein rings originally defined by V. Barucci and R. Fr\"oberg in 1997. What has dominated the…
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.
The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are…
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…
In this paper we study generalized Gorenstein Arf rings; a class of one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and…
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…
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…
In this paper we explore which part of the ideal lattice of a general ring is parametrized by its Cuntz semigroup $\mathrm{S}(R)$ and its ambient semigroup $\Lambda(R)$. We identify these classes of ideals as the quasipure ideals (a…
This note proves that if S is an unramified regular local ring and I and J are proper ideals of height at least two, then S/IJ is never Gorenstein.
In this paper, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein.…
The question of when the Rees algebra ${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring is explored, where $R$ is a two-dimensional regular local ring and $I$ a contracted ideal of $R$. It is known that…