Kamran Divaani-Aazar, Amir Mafi
Let $R$ be a commutative Noetherian ring. It is shown that $R$ is Artinian if and only if every $R$-module is good, if and only if every $R$-module is representable. As a result, it follows that every nonzero submodule of any representable $R$-module is representable if and only if $R$ is Artinian. This provides an answer to a question which is investigated in [{\bf 1}].
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We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian complete intersections have the strong Lefscehtz property.
On j-Artinian Modules Over Commutative Rings
本文研究了交换环上j-阿廷模块的性质，提出j-阿廷模块的定义，并建立了阿基次定理的类比。
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.
We study the conjecture that if $I,J$ are $μ-$primary in a regular local ring $(R,μ)$ with dim$(R)=n$, then $\frac{I \cap J}{IJ} \cong \Tor_1(R/I, R/J)$ needs at least $n$ generators, and a related conjecture about the number of generators of the first Koszul homology module of an Artinian local ring $(A,m)$. In this manuscript, we focus our attention on the complete intersection defect of the Artinian ring and its quotient by the Koszul elements. We prove that the number of generators of the first Koszul homology module of $x_1,...,x_n \in m$ on an Artinian local ring $(A,m)$ is at least $n+ …
In this paper, we make the notion of approximating an Artinian local ring by a Gorenstein Artin local ring precise using the concept of Gorenstein colength. We also answer the question as to when the Gorenstein colength is at most two.
An $R$-module $M$ is Hopfian (co-Hopfian) if any epic (monic) endomorphism of $M$ is an automorphism. If $R$ is commutative Noetherian, we characterize the co-Hopfian injective $R$-modules, and the Hopfian injectives in the case that $R$ is also reduced. For a commutative Artinian principal ideal ring, we show that $M$ is Hopfian (co-Hopfian) if and only if $M$ is finitely generated if and only if its injective envelope $E(M)$ is Hopfian (co-Hopfian) if and only if $E(M)$ is finitely generated. We identify the obstacle to generalizing this result to arbitrary Artinian principal ideal rings.
In this paper we give an effective characterization of Hilbert functions and polynomials of standard algebras over an Artinian equicharacteristic local ring; the cohomological properties of such algebras are also studied. We describe algorithms to check the admissibility of a given function or polynomial as a Hilbert function or polynomial, and to produce a standard algebra with a given Hilbert function.
We define a notion of compressed local Artinian ring that does not require the ring to contain a field. Let $(R,\mathfrak m)$ be a compressed local Artinian ring with odd top socle degree $s$, at least five, and $\operatorname{socle}(R)\cap \mathfrak m^{s-1}=\mathfrak m^s$. We prove that the Poincaré series of all finitely generated modules over $R$ are rational, sharing a common denominator, and that there is a Golod homomorphism from a complete intersection onto $R$.
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 that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.
Characterizing $S$-Artinianness by uniformity
本文通过一致性条件刻画 $u$-$S$-Artinian 模与环，证明其等价于 $u$-$S$-Noetherian、$S$-幂零根及半单商环，并给出区分三类环的例子。
Let $(R,\mathfrak{m})$ be a commutative local noetherian ring. For an artinian $R$-module $M$, we show the equality $$\mathrm{cosupp}_RM=\mathrm{Cosupp}_RM$$ using the semi-discrete linearly compactness of $R$-module $\mathrm{Hom}_R(R_\mathfrak{p},M)$ where $\mathfrak{p}$ is a prime ideal of $R$.
We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.
On the iso-Artinianness of one-dimensional Noetherian rings
本文完整刻画了一维Noether环的iso-Artinian性质，给出四个等价条件，并解决了Daneshvar和Divaani-Aazar提出的开放问题。
Bounds and MacWilliams Identities for codes over Artinian rings
该研究针对局部Artinian交换环上的线性编码，利用代数不变量拓展了码型与对偶自由秩的关系，给出汉明距离新上界与MDS码对偶保持条件，并提出新重量计数多项式证明了Frobenius环上的MacWilliams恒等式。
Betti numbers for connected sums of graded Gorenstein artinian algebras
本文确定了分次阿廷Gorenstein代数连通和的分次Betti数，并证明加倍代数的连通和等价于纤维积环的加倍，从而将连通和构造与加倍构造联系起来。
A Study of Good and Bad Artinian Gorenstein local Rings
本文研究了Artinian Gorenstein局部环的良坏性质，证明了特定条件下此类环为良好，并给出了不良环的例子。
The basic sequence of a homogeneous ideal $I\sset R=k[\seq{x}{1}{r}]$ defining an Artinian graded ring $A=R/I$ not having the weak Lefschetz property has the property that the first term of the last part is less than the last term of the penultimate part. For a general linear form $\ell$ in $\seq{x}{1}{r}$, this fact affects in a certain way the behavior of the $r-1$ square matrices in $k[\ell]$ which represent the multiplications of the elements of $A$ by $\seq{x}{1}{r-1}$ through a minimal free presentation of $A$ over $k[\ell]$. Taking advantage of it, we consider some modules over an algeb…
Let $R$ be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring $R[x,f]$ (associated to $R$ and the Frobenius homomorphism $f$, in the indeterminate $x$) that is both $x$-torsion and Artinian over $R$.
Tor and Ext vanishing results for commutative Artinian rings
该研究针对交换阿廷环领域的三个猜想，通过构造特定的代数与模，对Avramov等人、Huneke等人的猜想给出反例，回答了Dao的问题，推进了Tachikawa猜想的相关研究。
nil-$M$-Noetherian and nil-$M$-Artinian modules
本文在交换环模论中引入 nil-$M$-Noetherian 与 nil-$M$-Artinian 模概念，推广经典 Noetherian 与 Artinian 模，并建立相关代数概念的 nil 版本。
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 algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gor_l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory Art_l is companionable, with model-companion Gor_l.
Gerstenhaber showed in 1961 that any commuting pair of n x n matrices over a field k generates a k-algebra A of k-dimension \leq n. A well-known example shows that the corresponding statement for 4 matrices is false. The question for 3 matrices is open. Gerstenhaber's result can be looked at as a statement about the relation between the length of a 2-generator finite-dimensional commutative k-algebra A and the lengths of faithful A-modules. Wadsworth generalized this result to a larger class of commutative rings than those generated by two elements over a field. We recover his generalization, …
Let V be a finite-dimensional vector space over a field k and let W be a 1-dimensional k-vector space. Let < , >: V x V \to W be a symmetric bilinear form. Then < , > is called anisotropic if for all nonzero v \in V we have <v,v> \neq 0. Motivated by a problem in algebraic number theory, we come up with a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that one can check if a form is anisotropic by checki…
In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk Theorem, which deals with the case that R is a field (of any characteristic). Here we will show that for tameness over an Artinian ring, the Q-algebra assumption is really needed: we will give, for local Artinian rings with square-zero principal maximal ideal, a complete description of the tame automorphism subgroup. This will lead to an example of a non-tame aut…
Ascent and descent of Artinian module structures under flat base changes
本文研究交换 Noether 局部环间平坦局部同态下 Artinian 模结构的上升与下降，刻画 $A\otimes_RS$ 的结构，并用于判定某些局部上同调模的 Artinian 性。
Fix a pair of positive integers d and n. We create a ring R and a complex G of R-modules with the following universal property. Let P be a polynomial ring in d variables over a field and let I be a grade d Gorenstein ideal in P which is generated by homogeneous forms of degree n. If the resolution of P/I by free P-modules is linear, then there exists a ring homomorphism from R to P such that P tensor G is a minimal homogeneous resolution of P/I by free P-modules. Our construction is coordinate free.
This paper centers around Artinianness of the local cohomology of $ZD$-modules. Let $\fa$ be an ideal of a commutative Noetherian ring $R$. The notion of $\fa$-relative Goldie dimension of an $R$-module $M$, as a generalization of that of Goldie dimension is presented. Let $M$ be a $ZD$-module such that $\fa$-relative Goldie dimension of any quotient of $M$ is finite. It is shown that if $\dim R/\fa=0$, then the local cohomology modules $H^i_{\fa}(M)$ are Artinian. Also, it is proved that if $d=\dim M$ is finite, then $H^d_{\fa}(M)$ is Artinian, for any ideal $\fa$ of $R$ . These results exten…
We study two subcategories of the category of artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring.
Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\fa$ and $\fb$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\fa, \fb$, $\fa\cap\fb$ and $\fa+ \fb$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\fa$ such that all local cohomology modules of $M$ with respect to $\fa$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$, any maximal element $\fq$ of the non-empty set of ideals $\{\fa$ : $\H_\fa^i(M)$ is not artinian for some $i$, $i\geq r$$\}$…
Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors Tor_i^R(L,-) and Ext^i_R(L,-). For instance, we show the following: (a) if L is artinian and L' is noetherian, then Hom_R(L,L') has finite length; (b) if L and L' are artinian, then the tensor product L \otimes_R L' has finite length; (c) if L and L' are artinian, then Tor_i^R(L,L') is artinian, and Ext^i_R(L,L') is noetherian over the completion \hat R; and (d) if L is artinian and L' is Matlis reflexive, then Ext^i_R(L,L'), Ext^i_R(L',L), and Tor_i^R(L,L') are Matlis ref…
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of…
Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules Ext^i_R(L,L') and Tor_i^R(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that Ext^i_R(M,M'), Ext^i_R(M',M), and Tor_i^R(M,M') are Matlis reflexive over R for all i\geq 0 and that Ext^i_R(M,M')^\vee\cong Tor_i^R(M,M'^\vee) and Ext^i_R(M',M)^\vee\cong…
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified version of the Euclidean algorithm. Using the same strategy, we show how the reduced resultant and a pair of Bézout coefficient can be computed. Particular attention is devoted to the special case of $\mathbf{Z}/n\mathbf{Z}$, where we perform a detailed analysis of the asymptotic cost of the algorithm. Finally, we illustrate how the algorithms can be exploited …
Betti Numbers for Modules Over Artinian Local Rings
本文引入数值不变量 $γ_I(M)$，在 Tor 消失条件下联系其与 Betti 数，建立 $I$-自由模的自由性判据，并用于推出典范模 Betti 数条件下环的 Gorenstein 性质。
Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this article, we examine the question of whether an arbitrary top local cohomology module, $\operatorname{H}^{\operatorname{cd}(I,M)}_I(M)$, is Artinian, or not. Several results related to this question are obtained; in particular, we prove that over a Noetherian local unique factorization domain $R$ of dimension three, for a finitely generated faithful module $M$, a top local cohomology module is Artinian if and only if $\operatorname{cd}(I,M)= 3$.
Let $(R,\mathfrak m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M$ a weakly finite or a coatomic $R$-module of dimension $n$. In this article, we resolve the Artinianness and non-Artinianness of top local cohomology modules, $H^{cd(I,M)}_I(M)$, in all cases except in the case $cd(I,M)=n-1$ and $dim(R/{I+Ann(M)})>1$ for which we have some sorter results under certain conditions.
Let $R$ be a commutative ring and $M$ a non-zero $R$-module. We introduce the class of \emph{pseudo strongly hollow submodules} (\emph{PS-hollow submodules}, for short) of $M$. Inspired by the theory of modules with \emph{secondary representations}, we investigate modules which can be written as \emph{finite} sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of \emph{minimal} PS-hollow strongly representations of modules over Artinian rings.
Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian module over semi-local complete ring $R$ and $j$ is an integer such that $H_i^{\fa}(A)$ is Artinian for all $i<j$, then the set $\Coass_R(H_j^{\fa}(A))$ is finite. We also introduce the notion of the $n$th Artinianness dimension $g_n^\fa(A)=\inf\{g^{\fa R_{\fp}}(^\fp A): \fp\in\Spec(R) \ \ \text{and} \ \ \dim R/\fp\geq n\}$, for all $n\in\mathbb{N}_{0}$ and pr…
Telescope conjecture for quiver representations over artinian rings
本文证明了在艺术环上有限有向图表示的导出范畴中的准射线猜想，并扩展到交换完美环的非noetherian背景下。
Given positive integers e and s we consider Gorenstein Artinian local rings R of embedding dimension e whose maximal ideal $\mathfrak{m}$ satisfies $\mathfrak{m}^s\ne 0=\mathfrak{m}^{s+1}$. We say that R is a compressed Gorenstein local ring when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If $s\ne 3$, we prove that the Poincare series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends on…
Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.
The transfer of Artinian and Cohen-Macaulay properties under module-finite extensions
本文研究交换 Noether 局部环的模有限扩张下 Artinian 模结构与附着素理想的传递，并证明 R 为 Cohen-Macaulay 商环当且仅当 S 亦然，进而刻画 Nagata 理想化结构。
Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. In this paper we introduce some criterions on Artinianness of formal local cohomology, in particular vanishing and finiteness of local cohomology modules. We find out the lower and upper bound for Artinianness of formal local cohomology
The local-global principle for the artinianness dimensions
本文针对交换 noetherian 环上的 artinian 性维数，证明了在余伴素理想集合有限条件下的局部—整体等式，并研究了相应有限余伴素理想模类。
Let $R$ be a commutative ring and let $U(R)$ be multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, $Γ(R, G, S)$, corresponding to a multiplicative subgroup $G$ of $U(R)$ and a non-empty subset $S$ of $G$ with $S^{-1}=\{s^{-1} \mid s\in S\}\subseteq S$, as the graph with vertex set $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $s\in S$ such that $x+sy \in G$. In this paper, we characterize all Artinian rings $R$ whose $Γ(R,U(R), S)$ is projective. This leads to determine all Artinia…
We establish a link between trace modules and rigidity in modules over Noetherian rings. Using the theory of trace ideals we make partial progress on a question of Dao, and on the Auslander-Reiten conjecture over Artinian Gorenstein rings.
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax $R$-module $M$ of krull dimension less than 3, with respect to $\mathcal{S}$. There are some results on cofiniteness and artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module $M$ of finite krull dimension and an integer $n\in\mathbb{N}$, if $\lc^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then $\lc^{i}_{I,J}(M)/\fa^{j}\lc^{i…
Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs
该文研究含单位元交换环的素理想和图的图论性质与环代数性质的关联，分类了交叉帽数至多2、亏格为1的非局部交换Artin环，证明不存在素理想和图为射影平面的这类环。
We consider vanishing of Ext and Tor, especially over Artinian rings. In particular, we prove the Auslander-Reiten conjecture for all commutative local rings in which the cube of the maximal ideal is zero.
A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative description of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property, thus provid…
Restricted minimum condition in reduced commutative rings
该研究在交换环领域探讨限制极小（RM）条件，将Noether整环上RM条件等价于Krull维数≤1的结论推广至约化Noether环，证明CDR整环是RM环并给出Dedekind整环新刻画，还得到多项式环为RM环的充要条件，同时研究了RM环与UFD的关联。
On rings whose prime ideal sum graphs are line graphs
该研究针对含单位元的交换环，围绕素理想和图的结构性质展开，刻画了素理想和图为线图及线图补图的所有交换Artinian环，丰富了环与图交叉领域的结构刻画结果。
Artinian Gorenstein algebras of embedding dimension four and socle degree three
本文证明了在特征零代数闭域的多项式环中，特定Gorenstein齐次理想可由三级完美理想通过加倍构造，并给出了相应的分次极小自由分解的完整描述。
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。
Let GradAlg(H) be the scheme parameterizing graded quotients of R=k[x_0,...,x_n] with Hilbert function H (it is a subscheme of the Hilbert scheme of P^n if we restrict to quotients of positive dimension, see definition below). A graded quotient A=R/I of codimension c is called standard determinantal if the ideal I can be generated by the t by t minors of a homogeneous t by (t+c-1) matrix (f_{ij}). Given integers a_0\le a_1\le ...\le a_{t+c-2} and b_1\le ...\le b_t, we denote by W_s(\underline{b};\underline{a}) the stratum of GradAlg(H) of determinantal rings where f_{ij} \in R are homogeneous …
We establish basic results on subrings of finite commutative rings and closely related rings. Among other applications we calculate the number of maximal subrings of a finite commutative local ring.
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. We prove that if $R$ is local with maximal ideal $\fm$ and $ M\otimes_R N$ is of finite length then $H_{\fm}^t(M,N)$ is of finite length for all $t\geq 0$ and $l_R(H_{\fm}^t (M,N))\leq \sum_{i=0}^t l_R(\Ext_R^i(M,H_{\fm}^{t-i}(N)))$. This yields, $l_R(H_{\fm}^t(M,N))=l_R(\Ext_R^t(M,N))$. Additionally, we show that $\Ext_R^i(R/{\fa},N)$ is Artinian for all $ i\leq t$ if and only if $H_{\fa}^i(M,N)$ is Artinian for all $i\leq t$. Moreover, we show tha…
Gerko proves that if an artinian local ring $(R,\mathfrak{m}_R)$ possesses a sequence of strongly Tor-independent modules of length $n$, then $\mathfrak{m}_R^n\neq 0$. This generalizes readily to Cohen-Macaulay rings. We present a version of this result for non-Cohen-Macaulay rings.
Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with the vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We investigate commutative rings $R$ whose annihilating-ideal graphs have positive genus $γ(\Bbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $γ(\Bbb{AG}(R))<\infty$, then $R$ has finitely many ideals or $(R,\mathfrak{m})$ is a Gorenstein ring with maximal ideal $\mat…
In this paper, we introduce and study the $S$-weak global dimension $S$-w.gl.dim$(R)$ of a commutative ring $R$ for some multiplicative subset $S$ of $R$. Moreover, commutative rings with $S$-weak global dimension at most $1$ are studied. Finally, we investigated the $S$-weak global dimension of factor rings and polynomial rings.
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