相关论文: The Gorenstein Colength of an Artinian Local Ring
Given an Artinian local ring $R$, we define its Gorenstein colength $g(R)$ to measure how closely we can approximate $R$ by a Gorenstein Artin local ring. In this paper, we show that $R = T/I$ satisfies the inequality $g(R) \leq…
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…
The class of all Artinian local rings of length at most l is A_2-elementary, axiomatised by a finite set of axioms Art_l. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are…
We introduce classes of rings which are close to being Gorenstein. These rings arise naturally as specializations of rings of countable CM type. We study these rings in detail, and along the way generalize an old result of Teter which…
This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once $R$ is an almost Gorenstein graded ring, the localization $R_M$ of $R$ at the graded maximal ideal $M$ is almost…
Gorenstein homological algebra is a kind of relative homological algebra which has been developed to a high level since more than four decades. In this report we review the basic theory of Gorenstein homological algebra of artin algebras.…
Let $(R, \mathfrak{m}) $ be a Gorenstein local ring of dimension $d > 0$ and let $I$ be an ideal of $R$ such that $(0) \ne I \subsetneq R$ and $R/I$ is a Cohen-Macaulay ring of dimension $d$. There is given a complete answer to the question…
The main achievement of this paper is to provide a structure theorem for Artinian, Gorenstein local rings with the property that the square of the maximal ideal is generated by two elements. The moduli problem for this class of local…
In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings.…
The notion of $2$-almost Gorenstein ring is a generalization of the notion of almost Gorenstein ring in terms of Sally modules of canonical ideals. In this paper, we deal with two different topics related to $2$-almost Gorenstein rings. The…
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 notion of generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. In this article, there is given a characterization of GGL rings in terms of their canonical ideals and related…
The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fr\"oberg for one-dimensional Noetherian local rings which are analytically unramified has been generalized by S. Goto, N. Matsuoka and T. T. Phuong to…
In this paper, after giving a criterion for a Noetherian local ring to be quasi-Gorenstein, we obtain some sufficient conditions for a quasi- Gorenstein ring to be Gorenstein. In the course, we provide a slight generalization of a theorem…
The notion of 2-almost Gorenstein local ring (2-AGL ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of…
The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified.…
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 commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields…
We investigate when a commutative ring spectrum $R$ satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein…
In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attach prime ideals of certain local cohomology modules.