Euler class group of a Noetherian ring
M. K. Keshari
本文研究了由S.M. Bhatwadekar和Raja Sridharan。这是我2001年的哲学硕士论文
We will study the Euler class group of a commutative Noetherian ring $A$ of dimension $d$, developed by S.M. Bhatwadekar and Raja Sridharan. This is my M.Phil thesis from 2001.
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Another definition of Euler class group of a Noetherian ring
给出了诺特环的Euler类群的另一种定义。
In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if and only if $R[x]$ is Nil$_{\ast}$-Noetherian, if and only if $R[[x]]$ is Nil$_{\ast}$-Noetherian. Then we discuss some Nil$_{\ast}$-Noetherian properties on idealizations and bi-amalgamated algebras. Finally, we give the Cartan-Eilenberg-Bass Theorem for Nil$_{\ast}$-Noetherian rings in terms of Nil$_{\ast}$-injective modules and Nil$_{\ast}$-FP-injective …
In this paper, we introduce a new notion, called the integral dimension, for noetherian rings. It can be regarded as the weak Briancon-Skoda numbers of rings. The point is that every noetherian local ring has finite integral dimension.
For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $σ$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally $σ$-noetherian ring, then $σ$ is of finite type, and that totally $σ$-noetherian is a local property.
We consider trace ideals in Noetherian rings and focus our attention to one-dimensional analytically irreducible local rings. For such rings we classify those Gorenstein rings which admit only a finite number of trace ideals.
On the notion of Cohen-Macaulayness for non Noetherian rings
Noether Cohen-Macaulay环的刻画在文献中有很多。如果我们放弃诺特假设，这些特征就不再是等价的。本文的目的是提供一些比较这些特征在非诺特的情况下。为了解决Glaz提出的一个猜想，在非诺特环的情形下，我们推广了Hochster-Eagon关于不变环的Cohen-Macaulay性的结果.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. Then $R$ is called a uniformly $S$-Noetherian ($u$-$S$-Noetherian for abbreviation) ring provided there exists an element $s\in S$ such that for any ideal $I$ of $R$, $sI \subseteq K$ for some finitely generated sub-ideal $K$ of $I$. We give the Eakin-Nagata-Formanek Theorem for $u$-$S$-Noetherian rings. Besides, the $u$-$S$-Noetherian properties on several ring constructions are given. The notion of $u$-$S$-injective modules is also introduced and studied. Finally, we obtain the Cartan-Eilenberg-Bass Theorem for uniformly $S$-Noetheria…
Non-Noetherian Cohen-Macaulay rings
本文研究了有单位元的交换环的一个性质，它是每个凝聚正则环所具有的，并且等价于诺特环的Cohen-Macaulay性质.我们研究的情况下，环的扩展（各种类型）和环的不变量这个属性的行为。
On S-J-Noetherian Rings
本文提出S-J-Noetherian环的概念，推广了J-Noetherian环和S-Noetherian环，并证明了Cohen型定理及S-素分解的存在性。
Let R be a commutative ring with identity. We investigate some ring-theoretic properties of weakly Laskerian R-modules. Our results indicate that weakly Laskerian rings behave as Noetherian ones in many respects. However, we provide some examples to illustrate the strange behavior of these rings in some other respects.
We investigate flat maps where the source or target is a Noetherian ring, giving necessary and/or sufficient conditions on a ring for such maps to exist. Along the way, we develop some general facts about flat ring maps, and exhibit many examples, including a new class of zero-dimensional local rings.
In this note, we give the Cohen type theorem for uniformly $S$-Noetherian modules and the Eakin-Nagata type theorem for uniformly $S$-Noetherian rings. We also solve an open question proposed by Kim and Lim [5,Question 4.10].
In this note, we study the Cohen-Macaulayness of non-Noetherian rings. We show that Hochster's celebrated theorem that a finitely generated normal semigroup ring is Cohen-Macaulay does not extend to non-Noetherian rings. We also show that for any valuation domain $V$ of finite Krull dimension, $V[x]$ is Cohen-Macaulay in the sense of Hamilton-Marley.
We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple module as subquotient. We deduce in particular a characterization of uniserial modules over commutative noetherian rings.
Cohen, Levitzki, Hilbert Basis, and Lasker-Noether Theorems for Nil-S-Noetherian Rings
引入Nil-S-Noetherian环，推广S-Noetherian环和Nil*-Noetherian环，建立Cohen定理、Levitzki定理、Hilbert基定理和Lasker-Noether定理的广义版本。
On Z-Noetherian Rings
本文引入$Z$-诺特环的概念，研究其内在刻画、稳定性及相关定理，证明诺特环的部分基本性质可推广至该环类。
Projectively full ideals in Noetherian rings, a survey
讨论了诺特环中理想的射影等价和投射满理想的存在性与不存在性。我们描述的连接与里斯估值和里斯整数的理想，并考虑是否可以通过传递到一个整扩张环的改进。
We show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure-injective resolution of the ring. As an application of this, we prove that every countable commutative noetherian ring has full cosupport. We also settle the comparison of cosupport and support of finitely generated modules over any commutative noetherian ring of finite Krull dimension. Finally, we give an example showing that the cosupport of a finitely generated module need not be a closed subset of $\operatorname{Spec}(R)$, providing a negative answer to a question of Sathe…
Let $R$ be a Noetherian ring. We prove that $R$ has global dimension at most two if, and only if, every prime ideal of $R$ is of linear type. Similarly, we show that $R$ has global dimension at most three if, and only if, every prime ideal of $R$ is syzygetic. As a consequence, one derives a characterization of these rings using the André-Quillen homology.
Almost Noetherian rings and modules
本文研究了近诺特环和模的性质，提出了多个重要定理，并在特定条件下解决了相关问题。
Let $D$ be a 2-dimensional regular local ring and let $Q(D)$ denote the quadratic tree of 2-dimensional regular local overrings of $D$. We examine the Noetherian rings that are intersections of rings in $Q(D)$. To do so, we describe the desingularization of projective models over $D$ both algebraically in terms of the saturation of complete ideals and order-theoretically in terms of the quadratic tree $Q(D)$.
Let $(A,\mathfrak{m}, k=A/\mathfrak{m})$ be a noetherian local ring. Then it is equivalent $n = \dim A = \dim_k \mathfrak{m}/\mathfrak{m}^2$ and $\mathrm{Tor}^A_i(k,k) = 0$ for all $i \gg 0$. The article gives a proof with the change-of-ring spectral sequence in the derived category form.
Derivations, generic formal fibers and bad Noetherian rings
我们考虑一个循环的想法，涉及微分代数，局部诺特环，和他们的一般形式纤维。把这些概念联系起来，就产生了我们所说的环$S$的“扭曲”子环$R$。每个这样的子环$R$作为取$S$ -模块$K$中的值的导出的拉回而出现。扭曲关系被证明是Nagata理想化的一种反转：理想化将$S$扩展为更大的环$S \star K$，扭曲产生了$S$的一个子环，它的行为非常像环$S \star K$。以这种方式产生的环表现出病态的特征，例如不能具有有限的归一化，但尽管如此，它们是相当易处理的，并且在概念上（如果不是实际上）容易定位，并且以这种方式提供了丰富但可管理的非标准诺特环的来源。支持这种结构的理论涉及到诺特和非诺特交换环理论的广泛应用，以及微分代数。
In this paper, we consider the openness of the P-locus of a finitely generated module over a commutative noetherian ring in the case where P is each of the properties FID, Gor, CM, MCM, (S_n), and (T_n). One of the main results asserts that FID-loci over an acceptable ring are open. We give a module version of the Nagata criterion, and prove that it holds for all of the aforementioned properties.
Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.
We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension or finite injective dimension.
Let A be a Noetherian ring and B be a finitely generated A-algebra. Denote by A' the integral closure of A in B. We give necessary and sufficient conditions for prime ideals to be in Ass_{A}(B/A') and Ass_{A'}(B/A') generalizing and strengthening classical results for rings of special type.
We construct explicitly a resolution of a fan algebra of principal ideals over a Noetherian ring for the case when the fan is a proper rational cone in the plane. Under some mild conditions on the initial data, we show that this resolution is minimal.
Valuation rings and perfectoid rings are examples of (usually non-noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck's definition of formal smoothness as in the noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the noetherian to the…
On generalized fraction and power series properties of $\mathcal{S}$-Noetherian rings
本文研究了 $\mathcal{S}$-Noetherian 环的广义分数与幂级数性质，并具体探讨了相关文献中提出的两个问题。
Let $R$ be a complete regular local ring with an algebraically closed residue field and let $A$ be a Noetherian $R$-subalgebra of the polynomial ring $R[X]$. It has been shown in \cite{DO2} that if $\dim R=1$, then $A$ is necessarily finitely generated over $R$. In this paper, we give necessary and sufficient conditions for $A$ to be finitely generated over $R$ when $\dim R=2$ and present an example of a Noetherian normal non-finitely generated $R$-subalgebra of $R[X]$ over $R= {\mathbb C}[[u, v]]$.
Mixed Multiplicities of Multi-Graded Algebras Over Noetherian Local Rings
设S是Noether局部局部环A上的标准分次代数.本文首先用Hilbert-Samuel重数来表示S的混合重数，将S的混合重数解释为S的商模的Hilbert-Samuel重数。作为应用，我们得到了理想的混合重数的公式，涵盖了Trung-Verma 2007 [TV]的主要结果。
Some remarks on $u$-$S$-Noetherian and $u$-$S$-coherent rings
本文研究了$u$-$S$-Noetherian和$u$-$S$-coherent环的特征化问题，并对Bouziri提出的问题给出了否定回答。
Rings with uniformly S-w-Noetherian spectrum
本文引入一致S-w-Noetherian谱环的概念，给出其多个等价刻画，包括理想升链稳定性、理想根有限性、多项式环相关谱性质，并得到经典Noetherian谱情形的新结论。
Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.
We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity $R$ having at least one finitely generated ideal $I$ such that the quotient ring is again finitely generated, and $R$ is $I-$adically complete. Moreover, we offer as a direct corollary a new elementary proof of the fact that if a ring is Noetherian then the corresponding ring of formal power series in finitely many variables is Noetherian. In addition, we give a counterexample showing that the `completion' condition cannot be avoided on the former theorem. Lastly, we give an eleme…
On (uniformly) $S$-$w$-Noetherian rings and modules
该研究在环与乘法子集的代数框架下，引入（一致）$S$-$w$-Noetherian模、（一致）$S$-$w$-主理想模两类新概念，并给出其相关刻画性质，拓展了Noetherian环模相关理论。
In the derived category of modules over a commutative noetherian ring a complex $G$ is said to generate a complex $X$ if the latter can be obtained from the former by taking summands and finitely many cones. The number of cones required in this process is the generation time of $X$. In this paper we present some local to global type results for computing this invariant, and discuss applications.
The weakly Arf $(S_2)$-ification of a commutative Noetherian ring $R$ is considered to be a birational extension which is good next to the normalization. The weakly Arf property (WAP for short) of $R$ was introduced in 1971 by J. Lipman with his famous paper [12], and recently rediscovered by [4], being closely explored with further developments. The present paper aims at constructing, for a given Noetherian ring $R$ which satisfies certain mild conditions, the smallest module-finite birational extension of $R$ which satisfies WAP and the condition $(S_2)$ of Serre. We shall call this extensio…
The Noetherian Case of Bayart's Power-Series Question
针对Bayart1973年提出的问题，研究诺特环上形式幂级数环的唯一分解性，证明单变量形式幂级数环若为唯一分解整环，则双变量形式幂级数环也为该类整环，所用方法涉及诺特正规整环的除子理论。
Cancellation of projective modules over non-Noetherian rings
设R是维数为d的环，A是R[Y]或R[Y，Y^{-1}]中的一个.若P是秩为\geq d+1的投射A-模，满足一定条件，则证明了E（A \oplus P）传递作用于Um（A \oplus P）.当P是自由的，这个结果是由于Yengui（当A=R[Y]）和Abedelfatah（当A=R[Y，Y^{-1}]）。
Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors with supports in arbitrary subsets of Spec R, which is a natural generalization of right derived functors of section functors with supports in specialization-closed subsets. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for colocalization functors.
Let p be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies Tor^R_n(k(p),M) = 0 for some n \geq dim R_p, where k(p) is the residue field at p, then Tor^R_i(k(p),M) = 0 holds for all i \geq n. Similar rigidity results concerning Ext_R^*(k(p),M) are proved, and applications to the theory of homological dimensions are explored.
Conservation of the noetherianity by perfect transcendental field extensions
设$k$是特征$p>0$的完满域，$k(t)_{per}$是$k(t)$的完美闭包，$A$是$k$ -代数.我们刻画了环$A\otimes_k k(t)_{per}$是否是Noether环.证明了当$A$是$k$上$n$不定式的形式幂级数环时，环$A\otimes_k k(t)_{per}$是Noether环.
In this paper we prove the following generalization of a result of Hartshorne: Let $(S,\n)$ be a regular local ring of dimension $4$. Assume that $x,y,u,v$ is a regular system of parameters for $S$ and $a:=xu+yv$. Then for each finitely generated $S$-module $N$ with $\Supp N=V(aS)$ the socle of $H^2_{(u,v)S}(N)$ is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring $(R,\m)$, we characterize the class of all ideals $I$ of $R$ with the property that, for every finitely generated $R$-module $M$, the local cohomology modules $H^i_I(M)$ are $I$-cofinit…
The Japanese and universally Japanese properties for valuation rings and Prüfer domains
本文研究赋值环和Prüfer整环的Japanese与universally Japanese性质，证明二者等价，并推广Nagata经典结果至非Noetherian情形。
A partial converse ghost lemma for the derived category of a commutative noetherian ring
本文针对交换 Noether 环的有界导出范畴，证明可通过逐素理想局部化后的幽灵指数有限性来判定厚子范畴包含关系，并借助非负分次 Noether 同调 DG 代数建立逆协幽灵引理。
In this paper we study right $S$-Noetherian rings and modules, extending of notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right $S$-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain $S$-Noetherian modules with the following consequence in commutative algebra: If a module $M$ over a commutative ring is $S$-Noetherian with respect to a multiplicative set $S$ that contains no zero-divisors for $…
We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider the minimum among the number of generators of ideals $I$ such that either $I^2 = \mathfrak m^2$ or $I \supseteq \mathfrak m^2$ holds. We investigate subsequently the case that $R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by an ideal $I$ generated by homogeneous quadratic forms, and we compute these invariants. We devote specific attention to t…
Cohomological Cohen--Macaulayness in Non-Noetherian Rings
本文研究非诺特环的上同调Cohen--Macaulay性，证明局部有限维CCM环必为局部HMCM环，给出满足条件的环类实例，并建立特征p>0诺特环与其完备化的相关等价关系。
Colocalization and cotilting for commutative noetherian rings
对于交换诺特环R，研究了Mod-R和Mod-R_m中的倾斜模和余倾斜模之间的关系，其中m在R的极大谱上运行.对每个有限n，我们构造了n-余倾斜R-模C的（等价类）与n-余倾斜R_m-模（m \in mSpec R）的相容族F的（等价类）之间的1 - 1对应.它由赋值C导出|- >（C ^m; m \in mSpec R）其中C ^m是C在m处的共局部化，并且其逆F|- > \prod_{M \in F} M.我们构造了一个类似的对应n-倾斜模使用兼容家庭的本地化，但是，没有明确的公式为逆。
Polynomial Extensions and Localization of Non-Noetherian Cohen--Macaulay Rings
研究非诺特环中Hamilton-Marley意义下Cohen-Macaulay（HMCM）性质的多项式扩张保持性，以及无挠模的局部化现象，给出了稳定凝聚环有限弱整体维数下的保持性，并刻画了局部化保持无挠性的条件。
Power Series Rings over Zero-Dimensional Rings
本文研究零维基环上的幂级数环，提出保持平坦性且维数有限的版本，并将Krull高度定理和纯混合性推广至非诺特环情形。
Vanishing of the top local cohomology modules over Noetherian rings
设R是（不一定是局部的）Noether环，M是有限维d的R-生成模.设\fa是R的理想，\fM表示Supp_RH^ d_{\fa}（M）中所有素理想\fp的交.证明了H^ d_{\fa}（M）\simeq H^ d_{\fM}（M）/ \displaystyle{\sum_{n\in \mathbb{N}}}< \fM >（0：_{H^ d_{\fM}（M）} \fa^n），其中对于Artin R-模A，设< \fM >A= \cap_{n\in \mathbb{N}} \fM^n A.由此证明了对于R的所有理想\fa，只有1/2个非同构的顶局部上同调模H^ d_{\fa}（M）具有相同的支集.此外，我们建立了一个类似的Lichtenbaum-Hartshorne消失定理的环，不需要是本地的。
In this article, we generalize the well-known result that ideals of Noetherian polynomial rings have only finitely many initial ideals to the situation of ascending ideal chains in non-Noetherian polynomial rings. More precisely, we study ideal chains in the polynomial ring $R=K[x_{i,j}\,|\,1\leq i\leq c,j\in N]$ that are invariant under the action of the monoid $Inc(N)$ of strictly increasing functions on $N$, which acts on $R$ by shifting the second variable index. We show that for every such ideal chain, the number of initial ideal chains with respect to term orders on $R$ that are compatib…
Finite injective dimension over rings with Noetherian cohomology
研究了在上同调算子环的作用下具有诺特上同调的环.主要结果是一个标准的复杂的模在这样的环有有限内射维数。该准则通过去掉有限性条件，推广了已有的几个结果。特别地，我们证明了，对于一个模在环的诺特上同调，如果所有的高自扩张模消失，那么它必须有有限的内射维数。具有诺特上同调的环的例子包括交换完全交环和域上的有限维上交换霍普夫代数。
Noetherian rings of non-local rank
本文刻画了秩不等于其在极大理想处局部化秩上确界的诺特环，证明这类环必为有限个局部主 Artin 环与 Dedekind 整环的直积，并应用于 Artin 环上多项式环的局部秩计算。
The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can extend the concept of border bases over Noetherian rings whenever the corresponding residue class ring is finitely generated and free. In this paper we address the following problem: Can the concept of border basis over Noetherian rings exists for ideals when the corresponding residue class rings are finitely generated but need not necessarily be free modules?…
Examples of non-Noetherian domains inside power series rings
设R* 是Noether整环R的理想adic完备化，L是R* 的全商环的子域，使得L包含R.设A表示L与R* 的交集。整环A有时继承了R* 的一些好的性质，比如诺特性质。对于某些域L，可以使用与A相关联的R上的多项式环的嵌套并的局部化B来近似A;如果B是诺特环，则B = A。如果B不是诺特的，我们有时可以确定B的不是非线性生成的素理想。对每个正整数s，我们得到了一个三维局部唯一分解整环B，使得B的极大理想是2-生成的，B恰好有s个高度为2的素理想，B的每个高度为2的素理想都不是2-生成的，而B的所有其它素理想都是2-生成的.在这个例子中，我们研究了从Spec A到Spec B的映射。我们还提出了这个例子的推广到4维。我们描述了一个4维局部非Noether UFD B，使得B的极大理想是3-生成的，B的素理想Q恰好存在一个高度为3的，且该素理想Q不是3-生成的.我们考虑了Q是否是B的唯一不是可生成的素理想的问题，但还没有回答这个问题。
Uniformity in nonreduced rings via Noetherian operators
本文通过Noetherian微分算子证明Artin-Rees引理的微分版本，进而得到非约化环的若干均匀性结果。
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