On (strongly) Gorenstein (semi)hereditary rings
Najib Mahdou, Mohammed Tamekkante
本文引入并研究了Gorenstein同调维数小于或等于1的环，我们称之为Gorenstein（半）遗传环，特别是这类环的特殊情况，我们称之为强Gorenstein（半）遗传环.
In this paper, we introduce and study the rings of Gorenstein homological dimensions small or equal than 1, which we call Gorenstein (semi)hereditary rings, specially particular cases of these rings, which we call strongly Gorenstein (semi)hereditary rings.
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Gorenstein (semi)hereditary rings with respect to a semidualizing module
设$C$是一个半对偶模.我们首先研究了非线性生成的$G_C$ -投射模的性质。然后，相对于$C$，我们引入并研究了Gorenstein（弱）整体维数至多为1的环，称之为$C$ -Gorenstein（半）遗传环，并证明了每个$C$ -Gorenstein遗传环既是凝聚的，又是$C$ -Gorenstein半遗传的.
On torsion-free modules and semi-hereditary rings
本文研究了半遗传环的结构理论，重点探讨其与无挠模平坦性的关系，并考虑了Shimomoto关于Frobenius映射平坦性的问题。
Global Gorenstein dimensions of polynomial rings and of direct product of rings
本文将著名的Hilbert合冲定理推广到环的Gorenstein同调维数。研究了环的直积的Gorenstein同调维数，给出了有限Gorenstein维数和无限经典弱维数的非诺特环的例子.
Global Gorenstein dimensions of polynomial rings and of direct products of rings
在本文中，我们将著名的Hilbert合合定理推广到环的Gorenstein同调维数上。同时，我们还研究了环的直积的Gorenstein同调维数。我们的结果产生的例子，非诺特环的有限Gorenstein维数和无限的经典弱维数。
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 purposes are to determine all the Ulrich ideals in $2$-almost Gorenstein rings and to clarify the structure of minimal free resolutions of $2$-almost Gorenstein rings.
Gorenstein Von Neumann regular rings
本文研究了Gorenstein弱维数为零的环，称之为Gorenstein Von Neumann正则环。
For any commutative ring $A$ we introduce a generalization of $S$--artinian rings using a hereditary torsion theory $σ$ instead of a multiplicative closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $σ$--artinian ring, then $σ$ must be of finite type, and $A$ is totally $σ$--noetherian.
In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein and almost Gorenstein and to be non-level and almost Gorenstein in terms of the structure of the partially ordered set defining the Hibi ring. We also state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein in terms of the shape of the ladder.
When do pseudo-Gorenstein rings become Gorenstein?
本文研究伪Gorenstein环成为Gorenstein环的条件，证明在温和假设下伪Gorenstein的近乎Gorenstein分次整环必为Gorenstein，并阐明相关概念关系及给出构造拟Gorenstein环的方法。
Rings which are almost Gorenstein
我们介绍类环是接近Gorenstein。这些环是可数CM型环的特化而自然产生的。我们研究这些环的详细，并沿着这一道路推广了一个旧的结果特的特点是Artinian环的Gorenstein环模他们的socle。
Non-commutative resolutions for Segre products and Cohen-Macaulay rings of hereditary representation type
本文利用倾斜理论与丛倾斜理论研究遗传表示型Cohen-Macaulay环，证明Segre积上丛倾斜模或非交换crepant消解的存在性定理，构造多个Gorenstein环例子，并对一个4维非Gorenstein环给出刚性Cohen-Macaulay模的完全分类。
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 invariants.
We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In section 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Goren…
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 Nagata's idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a,b$; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a,b$. More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we …
Gorenstein global dimension of Semi-primary rings
本文主要研究半准素环的Gorenstein整体维数和弱维数。
The far-flung Gorenstein numerical semigroup rings of type 4
本文研究了类型4的远距离Gorenstein数值半群环的分类问题。
Rings over which all modules are strongly Gorenstein projective
本文的主要结果之一是刻画了其上所有模都是强Gorenstein投射的环。我们证明了这类环是著名的拟弗罗贝尼乌斯环的非常特殊的情形。我们给出的例子环上的所有模块是Gorenstein投射，但不一定强Gorenstein投射。
Gorenstein homology, relative pure homology and virtually Gorenstein rings
我们考虑以下问题：Gorenstein同调是X-纯同调，在这个意义上定义的沃菲尔德，一类X的模块？设GP表示Gorenstein投射模类.证明了有限克鲁尔维数的交换诺特环R上的Gorenstein同调是GP-纯同调的充要条件是R实质上是Gorenstein.
On Gorenstein global dimension in trivial ring extensions
本文比较了环$R$及其模$E$的平凡环扩张的Gorenstein同调维数.
In this paper we investigate the question of when the determinantal ring $R$ over a field $k$ is an almost Gorenstein local/graded ring in the sense of Goto, Takahashi, and the author. As a consequence of the main result, we see that if $R$ is a non-Gorenstein almost Gorenstein local/graded ring, then the ring $R$ has a minimal multiplicity.
Gorenstein dimensions in trivial ring extensions
本文证明了平凡环扩张的Gorenstein整体维数往往是无穷大的。此外，我们还研究了Gorenstein性质在一个环和它的平凡环扩张之间的转移。我们的结论与一个例子表明，在一般情况下，转移的概念Gorenstein投射模不进行拉回建设。
Computing families of Cohen-Macaulay and Gorenstein rings
刻画了由某些类型的凸体半群得到的Cohen-Macaulay环和Gorenstein环.给出了判定多边形半群或圆半群是Cohen-Macaulay/Gorenstein半群的数学方法。我们也给出了Cohen-Macaulay环和Gorenstein环的一些族。
On $n$-strongly Gorenstein rings
本文引入并研究了具有有限Gorenstein整体的交换环类的一个特殊子类（分别为，弱）维度。
$n$-strongly Gorenstein rings
本文引入并研究了有限Gorenstein整体维数的交换环类的一个特殊子类。
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.
Gorenstein Global Dimensions and Cotorsion Dimension of Rings
本文推广了交换环的经典同调维数的一个结果，利用环的整体余挠维数建立了交换环的Gorenstein整体维数的一个上界.我们使用这个结果来计算Gorenstein全球的一些特殊情况下平凡的扩展环和群环的维数。
How many ideals whose quotient rings are Gorenstein exist?
本文研究Gorenstein环中商环为Gorenstein的非主理想数量问题，证明对称数值半群环中此类分次理想个数等于半群导子，并给出完整列表。
Rings over which the class of Gorenstein flat modules is closed under extensions
环$R$称为左GF-闭的，如果所有Gorenstein平坦左$R$ -模的类在扩张下闭.左GF-闭环类严格包括右凝聚环类和有限弱维数环类。本文研究了左GF-闭环上的Gorenstein平坦维数。也就是说，我们推广的事实，所有Gorenstein平坦左模类是投射分解的右凝聚环的左GF-闭环。此外，我们还将右凝聚环上Gorenstein平坦左模的刻画（进而将左模的Gorenstein平坦维数的刻画）推广到左GF-闭环上。最后，利用环的直积，我们证明了如何构造一个既不是右凝聚环也不是有限弱维数的左GF-闭环。
A note on quasi-Gorenstein rings
本文在给出Noether局部环是拟Gorenstein环的一个判别准则后，得到了拟Gorenstein环是Gorenstein环的几个充分条件.在课程中，我们提供了埃文斯和格里菲斯定理的轻微推广。
First, second, and third change of rings theorems for the Gorenstein homological dimensions
出于他们的影响同调代数，改变环的结果一直是主题的几个有趣的工作在Gorenstein同调代数的诺特环。本文研究了任意环上Gorenstein维数的环变换定理。也就是说，通过使用的概念，强Gorenstein模，我们扩展了著名的第一，第二，第三个环的变化定理的经典投射和内射的尺寸Gorenstein投射和内射的尺寸，分别。本文建立的Gorenstein投射维数的每个结果都是诺特环上和有限生成模上的一个结果的推广。本文研究了任意环上Gorenstein维数的环变换定理。也就是说，通过使用的概念，强Gorenstein模，我们扩展了著名的第一，第二，第三个环的变化定理的经典投射和内射的尺寸Gorenstein投射和内射的尺寸，分别。本文建立的Gorenstein投射维数的每一个结果都是诺特环$R$上有限生成模$M$的$G$ -维数的推广.
Gorenstein weak dimension of a coherent power series rings
计算了交换环上的凝聚幂级数环的Gorenstein弱维数，并证明了一般情况下，$\gwd(R) \leq 1$并不意味着$R$是算术环.
$F$-stable submodules of top local cohomology modules of Gorenstein rings
本文应用G. Lyubeznik的$F$概念--有限模，以非常实际的方式描述Gorenstein环上某些顶级局部上同调模的某些零化子模。作为结果，我们得到了Gorenstein环的测试理想在正则环中的理想的明确描述。
Gorenstein injective filtrations over rings with dualizing complexes
本文将Gorenstein环上Gorenstein内射模的有限Gorenstein内射子模滤过结论，推广至带对偶复形的交换诺特环，并用Auslander范畴和截面函子实现了这类滤过。
We investigate the existence of ideals $I$ in a one-dimensional Gorenstein local ring $R$ satisfying $\mathrm{Ext}^{1}_{R}(I,I)=0$.
The Gorenstein Colength of an Artinian Local Ring
本文利用Gorenstein余长的概念，给出了Gorenstein Artin局部环逼近Artin局部环的精确概念.我们还回答了这个问题，当Gorenstein colength是最多两个。
Gorenstein global dimension of an amalgamated duplication of a coherent ring along an ideal
本文研究了凝聚环的\emph{amalgamated duplication}沿正则主理想的Gorenstein整体维数.
Koszulness of Gorenstein quadratic algebras of small socle degree is studied. In this note, we construct non-Koszul Gorenstein quadratic toric ring such that its socle degree is more than 3 by using stable set polytopes.
Almost Gorenstein standard graded rings and their localizations
本文证明了域上Cohen-Macaulay标准分次整环为几乎Gorenstein分次环，等价于其在分次极大理想处局部化的完备化为几乎Gorenstein局部环。
We show that $F$-purity deforms in local $\mathbb{Q}$-Gorenstein rings of prime characteristic $p>0$. Furthermore, we show that $F$-purity is $\mathfrak{m}$-adically stable in local Cohen-Macaulay $\mathbb{Q}$-Gorenstein rings.
In this paper, for the development of the study of almost Gorenstein graded rings, we discuss some relations between almost Gorensteinness of Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a Cohen--Macaulay homogeneous ring $R$, we give a sufficient condition for $R$ to be almost Gorenstein in terms of the $h$-vector of $R$ and we also characterize almost Gorenstein homogeneous domains with small socle degrees in terms of the $h$-vector of $R$. Moreover, we also provide the examples of almost Gorenstein homogeneous domains arising from lattice polytopes.
Rings over which all (finitely generated) strongly Gorenstein projective modules are projective
本文的主要目的是研究其上所有（强生成的）Gorenstein投射模都是投射模的环.我们认为这种适当的变化下的戒指，并给出各种例子的戒指和没有这种适当的。
Gorenstein Endomorphism Rings on Curve Singularities
本文研究曲线奇点上分式理想自同态环的 Gorenstein 性，利用理想稳定性与值半群理想条件给出刻画，并分类相关 Gorenstein 代数型曲线。
The Auslander-Bridger formula and the Gorenstein property for coherent rings
研究了Auslander和Bridger关于诺特环上的有限生成模的Gorenstein维数的概念，并在凝聚环上的有限表示模的背景下讨论了Gorenstein维数. Auslander-Bridger公式的一个推广的建立，并作为一个基石的发展理论的连贯Gorenstein环。
Almost Gorenstein rings
本文推广了Barucci和Fr{<$} \cite{BF}在局部环是解析非分歧的情形下给出的几乎Gorenstein环的概念，使它在局部环是解析分歧的情形下也同样有效.作为后续，当$\m$的自同态代数$\m : \m$是Gorenstein环时，完全一般地解决了问题，其中$\m$表示给定的1维Cohen-Macaulay局部环中的极大理想。结合理想化原理给出了几乎Gorenstein环的特征.实例进行了探讨。
Characterization of almost Gorenstein rings in terms of the trace ideal
本文用迹理想刻画一维几乎Gorenstein环，并应用于研究$\mathbb{Z}_2$-分次环的几乎Gorenstein性质。
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 give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Fröberg concerning Arf numerical semigroup rings.
A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.
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 rings $R$, $S$ of the same dimension $d>0$ over a regular local ring $T$ with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are $R$ and $S$. Besides, the other generalizations of Gorenstein properties are also explored.
Gorenstein injective dimension, Bass formula and Gorenstein rings
设$(R,\frak m, k)$是Noether局部环.众所周知，$R$是正则的当且仅当$k$的内射维数是有限的。本文证明了$R$是Gorenstein当且仅当$k$的Gorenstein内射维数是有限的.另一方面，对有限Gorenstein内射维数的有限生成模证明了Bass公式的一个推广形式.它还改进了Christensen的广义Bass公式（cf.“Gorenstein dimensions”，数学讲义第1747卷，Springer-Verlag，柏林，2000年）。
Remarks on almost Gorenstein rings
本文考察分次环与局部环几乎 Gorenstein 性质的传递关系，证明分次情形可推出极大理想局部化后的局部情形，并在一维分次整环的温和条件下给出逆命题，同时研究相关数值半群环的定义理想。
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 of when the idealization $A = R \ltimes I$ of $I$ over $R$ is an almost Gorenstein local ring.
Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the André-Quillen homology.
In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of nearly Gorensteinness, introduced by Herzog-Hibi-Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and …
We provide a characterization of the almost Gorenstein property of determinantal rings of a symmetric matrix of indeterminates over an infinite field. We give an explicit formula for ranks of the last two modules in the resolution of determinantal rings using Schur functors.
Poincaré series of modules over compressed Gorenstein local rings
给定正整数e和s，我们考虑嵌入维数为e的Gorenstein Artin局部环R，其极大理想$\mathfrak{m}$满足$\mathfrak{m}^s\ne 0=\mathfrak{m}^{s+1}$ .我们称R是压缩Gorenstein局部环，当它在这类环中具有最大长度时.已知一般Gorenstein Artin代数是压缩的。如果是$s\ne 3$，则证明了压缩Gorenstein局部环上所有有限生成模的Poincare级数都是有理的，并且有一个公分母.给出了分母的计算公式。当s是偶数时，这个公式只依赖于整数e和s。请注意，$s=3$的例子压缩Gorenstein局部环存在超越庞加莱级数，由于Bøgvad。
A class of transversal polymatroids with Gorenstein base ring
本文利用多面体几何，特别是刻画标准模的$Danilov-Stanley$定理，描述了具有Gorenstein基环的横截多拟阵。同时，我们还计算了这类横截多拟阵的基环的$a-invariant$和Hilbert级数。
Symmetry in the vanishing of Ext over Gorenstein rings
研究了局部Gorenstein环上有限生成模的Ext消失的对称性.特别地，我们定义了一类局部Gorenstein环，我们称之为AB环，并表明，对于AB环$R$上有限生成模$M$和$N$，对于所有$i >> 0$，$Ext^i_R(M,N)=0$当且仅当$Ext^i_R(N,M)=0$对于所有$i >> 0$。
A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring $R$ of regularity three is Koszul. In a previous paper, we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three which are not Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r…
When every Gorenstein projective (resp. flat) module is strongly Gorenstein projective (resp. flat)
在\cite{Ouarghi}中，作者讨论了其上所有模都是强Gorenstein投射的环.在本文中，我们感兴趣的是这一想法的扩展。我们将讨论每个Gorenstein投射（分别）的环。flat）模是强Gorenstein投射（或flat）模。我们的目的是给出满足这个条件的具有不同Gorenstein整体维数的环的例子。
Ideals defining Gorenstein rings are (almost) never products
本文证明了：若S是非分歧的正则局部环，且I和J是高度至少为2的真理想，则S/IJ决不是Gorenstein环.
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