Francois Couchot
It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings.
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We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $π$-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.
The classical Gelfand-Kirillov dimension for algebras over fields has been extended recently by J. Bell and J.J Zhang to algebras over commutative domains. However, the behavior of this new notion has not been enough investigated for the principal algebraic constructions as polynomial rings, matrix rings, localizations, filtered-graded rings, skew $PBW$ extensions, etc. In this paper, we present complete proofs of the computation of this more general dimension for the mentioned algebraic constructions for algebras over commutative domains. The Gelfand-Kirillov dimension for modules and the Gel…
Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.
We study the problem of existence of essential indecomposable submodules of direct sums of copies of the ring of quotients of Ore domains. We provide, for each Ore domain $D$, with at least three non-associate irreducibles, a lower bound for the supremum of all cardinals $κ$ such that the direct sum of $κ$ copies of the ring of quotients of $D$ contains indecomposable essential $D$-submodules, and a lower bound for the number of times this bound is attained up to isomorphisms. We provide examples illustrating these results.
Let T be a complete discrete valuation ring and $\hat{X}$ a smooth projective curve over $S=\spec(T)$ with closed fibre $X$. Denote by $F$ the function field of $\hat{X}$ and by $\hat{F}$ the completion of $F$ with respect to the discrete valuation defined by $X$, the closed fibre. In this paper, we construct indecomposable and noncrossed product division algebras over $F$. This is done by defining an index preserving group homomorphism $s:\br(\hat{F})'\to\br(F)'$, and using it to lift indecomposable and noncrossed product division algebras over $\hat{F}$.
Representation theory of the real Gelfand order and real Harish-Chandra modules for $\mathsf{SL}_2(\mathbbm{R})$
本文研究了实Harish-Chandra模对于SL₂(ℝ)群的主块，并将其与有限维模 over 所谓的实Gelfand顺序相关联，描述了几类对应的不可分解表示。
Rings whose indecomposable modules are pure-projective or pure-injective
本文刻画了所有不可分解模均为纯投射或纯内射的环类，证明了诺特局部交换环情形等价于阿廷赋值环或特定秩条件的离散赋值域，并将一般交换环情形归结为 clean 算术环及其局部化条件。
The Gelfand--Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand--Kirillov dimension of simple modules over certain simple rings of differential operators.
Centrally Essential Factor Rings and Subdirect Indecomposability
本文研究中心本质环的结构性质，证明了相关理想包含关系与环的局部性结论，刻画了两类次直不可约相关的中心本质环，并给出非交换实例，得到俄罗斯相关科研项目资助。
Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. In this paper we define one of the dimensions called couniserial dimension that measures how close a ring or module is to being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension like each module having such a dimension contains a uniform submodule and has finite uniform dimension, among others. Like all dimensions, this is…
A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3 (1969), 655-658] Dickson and Fuller had shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements then an indecomposable automorphism-invariant right $R$-module must be quasi-injective. In this paper we show that this result fails to hold if $\mathbb F$ is a field with two elements. Dickson and Fuller had further sho…
Remarks on undecidability of indecomposable polynomial rings
本文研究不可分解多项式环的不可判定性，证明算术可在任意变量集合的此类环中解释，并给出适用于该环类全体成员的统一替代证明。
Using the concept of ring of Gelfand range 1 we proved that a commutative Bezout domain is an elementary divisor ring iff it is a ring of Gelfand range 1. Obtained results give a solution of problem of elementary divisor rings for different classes of commutative Bezout domains, in particular, $PM^{*}$, local Gelfand domains and so on.
We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings, \cite[Theorem 5]{jacobi}. We show that this theorem is a consequence of the Gelfand-Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand-Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.
Endomorphism rings of simple modules and block decomposition
本文利用左右诺特半完全环不可分解性的特征刻画，通过Ext群研究单模，证明不可分解半完全环的单模基数及自同态环特征的一致性，并在拟Frobenius环情形下强化了结果。
Classifying Finitely Generated Indecomposable RA Loops
本文将有限RA环的分类结果推广至有限生成情形，并显式描述所有不可分解的有限生成RA环。
We consider Ore extensions of the form $T:=R[x;σ,δ]$ with $R$ a commutative integral domain that is finitely generated over a field $k$. We show that if $T$ has Gelfand-Kirillov dimension less than four then a prime ideal $P\in {\rm Spec}(T)$ is primitive if and only if $\{P\}$ is locally closed in ${\rm Spec}(T)$, if and only if the Goldie ring of quotients of $T/P$ has centre that is an algebraic extension of $k$. We also show that there are examples for which these equivalences do not all hold for $T$ of integer Gelfand-Kirillov dimension greater than or equal to $4$.
The lower transcendence degree, introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras, the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD-stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebr…
For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms be…
If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring.
On Artin algebras whose indecomposable modules are determined by composition factors
本文研究Artin代数中不可分解模由合成因子决定的性质，证实了该性质蕴含有限表示型的猜想在交换、遗传及根平方为零等特殊情形下成立。
It is shown that each almost maximal valuation ring R such that every indecomposable injective module is countably generated, satisfies the following condition (C): each fp-injective module is locally injective. The converse holds if R is a domain. Moreover, it is proved that a valuation ring that sastisfies this condition (C) is almost maximal.
Bongartz and Ringel proved that there is no gaps in the sequence of lengths of indecomposable modules for the finite-dimensional algebras over algebraically closed fields. The present paper mainly study this "no gaps" theorem for the bounded derived module category $D^b(A)$ of a gentle algebra $A$: if there is an indecomposable object in $D^b(A)$ of cohomological length $l>1$, then there exists an indecomposable with cohomological length $l-1$.
It is proven that every commutative semihereditary Bezout ring in which any regular element is Gelfand (adequate), is an elementary divisor ring.
In this paper, we study partial actions of groups on $R$-algebras, where $R$ is a commutative ring. We describe the partial actions of groups on the indecomposable algebras with enveloping actions. Then we work on algebras that can be decomposed as product of indecomposable algebras and we give a description of the partial actions of groups on these algebras in terms of global actions.
We classify indecomposable pure injective modules over domestic string algebras, verifying Ringel's conjecture on the structure of such modules.
For central simple finitely generated algebras of finite Gelfand-Kirillov dimension and for their division algebras upper bounds are obtained for the transcendence degree of their commutative subalgebras and subfields respectively. In the formulae for the upper bounds are used only two dimensions of the algebras: the Gelfand-Kirillov and the filter dimension. In the case of the ring of differential operators on a smooth irreducible affine algebraic variety the upper bounds are the exact upper bounds. A similar upper bound is given for the transcendence degree of isotropic subalgebras of strong…
Tensor Product and the Stable Green Ring of the Symmetric Group Algebra $F\mathfrak{S}_p$
研究对称群代数F S_p的不可约非投射模的张量积分解，证明其模投射模为半单，并计算Benson-Symonds不变量。
In this paper we introduce the semi-graded rings, which extend graded rings and skew PBW extensions. For this new type of non-commutative rings we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand-Kirillov dimension. We will extended the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre-Artin-Zhang-Verevkin theorem. Some examples are included at the end of the paper.
This article deals mostly with the following question: when is the classical ring of quotients of a commutative ring a ring of stable range 1? We introduce the concepts of a ring of (von Neumann) regular range 1, a ring of semihereditary range 1, a ring of regular range 1, a semihereditary local ring, a regular local ring. We find relationships between the introduced classes of rings and known ones, in particular, it is established that a commutative indecomposable almost clean ring is a regular local ring. A commutative any ring of idempotent regular range 1 is an almost clean ring. It is sho…
We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.
We introduce the notions of Strongly harmonic and Gelfand module, as a generalization of the well-known ring theoretic case. We prove some properties of these modules and we give a characterization via their lattice of submodules and their space of maximal submodules. It is also observed that, under some assumptions, the space of maximal submodules of a strongly harmonic module constitutes a compact Hausdorff space whose frame of open sets is isomorphic to the frame $Ψ(M)$ defined in [arXiv:1612.07407]. Finally, we mention some open questions that arose during this investigation.
Classification of Leavitt Path Algebras with Gelfand-Kirillov Dimension <4 up to Morita Equivalence
本文利用约化算法与 Serre 子范畴滤过，引入了有向图的加权 Hasse 图作为 Morita 不变量，并给出了 Gelfand-Kirillov 维数小于 4 的 Leavitt 路代数在 Morita 等价意义下的完备分类。
Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the $m$-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector representation. It is proved for O(n) $(n\geq 2)$ and for SO(n) $(n\geq 3)$ that there exist $m$--linear invariants with $m$ arbitrarily large that are indecomposable (i. e., not expressible as polynomials in invariants of lower degree). In fact, they are explicitly constructed for all possible …
Let $A$ be an Iwanaga-Gorenstein ring. Enomoto conjectured that a self-orthogonal $A$-module has finite projective dimension. We prove this conjecture for $A$ having the property that every indecomposable non-projective maximal Cohen-Macaulay module is periodic. This answers a question of Enomoto and shows the conjecture for monomial quiver algebras and hypersurface rings.
Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and …
Twisted generalized Weyl algebras (TGWAs) $A(R,σ,t)$ are defined over a base ring $R$ by parameters $σ$ and $t$, where $σ$ is an $n$-tuple of automorphisms, and $t$ is an $n$-tuple of elements in the center of $R$. We show that, for fixed $R$ and $σ$, there is a natural algebra map $A(R,σ,tt')\to A(R,σ,t)\otimes_R A(R,σ,t')$. This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all $t$) of the Grothendieck groups of the categories of weight modules for $A(R,σ,t)$. We give presentations of these Grothendieck rings for $n=1,2$, when $R=\mathbb{C}[z]…
We study the classical Köthe's problem, concerning the structure of non-commutative rings with the property that: ``every left module is a direct sum of cyclic modules". In 1934, Köthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring $R$ is called a ${\it left~Köthe~ring}$ if every left $R$-module is a direct sum of cyclic $R$-modules. In 1951, Cohen and Kaplansky proved that all commutative K{ö}the rings are Artinian principal ideal rings. During the years 1962 to 1965, Kawada solved the Köthe's problem for basic fnite-dimensional algebras…
Almost Commutative Terwilliger Algebras II: Strong Gelfand Pairs
本文研究由有限群强 Gelfand 对经 Schur 环构造的 Terwilliger 代数，并完全分类了其中几乎交换的情形。
The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$, \cite{JKS16}. Among the indecomposable modules in this category are the rank $1$ modules which are in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in ${\rm CM}(B_{k,n}) $ can be filtered by the…
Indecomposability of graded modules over a graded ring
本文证明了在 Henselian 局部环上的非负分次环及其有限型分次代数中，模的不可分解性在分次、局部化和完备化意义下等价，并给出了分次模分解的 Krull-Schmidt 唯一性定理及其在特征 p 环 FFRT 性质上的应用。
For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality of Ziegler spectra.
A ring $R$ is called left $k$-cyclic if every left $R$-module is a direct sum of indecomposable modules which are homomorphic image of $_{R}R^k$. In this paper, we give a characterization of left $k$-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe-Cohen-Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left $k$-cyclic ring. As a corollary, we show that $R$ is Morita equivalent to a basic left Köthe ring if and only if $R$ is an artinian left multiplicity-free top ring.
Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of nxn-matrices for n at least 3. The same obstruction applies to the Zariski, Stone, and Pierce spectra. The possibility of spectra in categories other than that of locales is briefly discussed.
We prove a tight connection between reflexive modules over a one-dimensional ring $R$ and its birational extensions that are self-dual as $R$-modules. Consequently, we show that a complete local reduced Arf ring has finitely many indecomposable reflexive modules up to isomorphism, which can be represented precisely by the local rings infinitely near it. We also characterize Arf rings by the property that any reflexive module is self-dual. We give applications on dimension of subcategories and existence of endomorphism rings of small global dimension (non-commutative desingularizations). Our re…
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