Let $Λ$ be a left and right noetherian ring and $\mod Λ$ the category of finitely generated left $Λ$-modules. In this paper we show the following results: (1) For a positive integer $k$, the condition that the subcategory of $\mod Λ$ consisting of $i$-torsionfree modules coincides with the subcategory of $\mod Λ$ consisting of $i$-syzygy modules for any $1\leq i \leq k$ is left-right symmetric. (2) If $Λ$ is an Auslander ring and $N$ is in $\mod Λ^{op}$ with $\grade N=k<\infty$, then $N$ is pure of grade $k$ if and only if $N$ can be embedded into a finite direct sum of copies of the $(k+1)$st term in a minimal injective resolution of $Λ$ as a right $Λ$-module. (3) Assume that both the left and right self-injective dimensions of $Λ$ are $k$. If $\grade {\rm Ext}_Λ^k(M, Λ)\geq k$ for any $M\in\mod Λ$ and $\grade {\rm Ext}_Λ^i(N, Λ)\geq i$ for any $N\in\mod Λ^{op}$ and $1\leq i \leq k-1$, then the socle of the last term in a minimal injective resolution of $Λ$ as a right $Λ$-module is non-zero.
Let $G$ be a group with neutral element $e$ and let $S=\bigoplus_{g \in G}S_g$ be a $G$-graded ring. A necessary condition for $S$ to be noetherian is that the principal component $S_e$ is noetherian. The following partial converse is well-known: If $S$ is strongly-graded and $G$ is a polycyclic-by-finite group, then $S_e$ being noetherian implies that $S$ is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain…
We generalize the tilting process by Happel, Reiten and Smalø to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones to all right noetherian rings.
We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancellative. We introduce the non-unital generalizations of graded analogues of noetherian and artinian rings, graded locally noetherian and graded l…
Two criteria are given for a ring to have a left Noetherian left quotient ring (this was an open problem since 70's). It is proved that each such ring has only finitely many maximal left denominator sets.
We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that the ring of differential operators is left Noetherian if the condition is satisfied. Moreover we give some other conditions for the ring of differential operators being left Noetherian. Finally we conject…
Whereas Holm proved that the ring of differential operators on a generic hyperplane arrangement is finitely generated as an algebra, the problem of its Noetherian properties is still open. In this article, after proving that the ring of differential operators on a central arrangement is right Noetherian if and only if it is left Noetherian, we prove that the ring of differential operators on a central 2-arrangement is Noetherian. In addition, we prove that its graded ring associated to the order filtration is not Noetherian when the number of the consistuent hyperplanes is greater than 1.
In this short note we study the links of certain prime ideals of a noetherian ring R. We first give the definition of a link krull symmetric noetherian ring R. We then prove theorem 9 that states that for any linked prime ideals P' and Q' of the polynomial ring R[X] where R is a link krull symmetric noetherian ring, if The prime ideal P' is extended then Q' is also an extended prime ideal of R[X]. An application of theorem 9 is then given in theorem 12 for the ring R[X] when R is assumed to be a fully bounded noetherian ring.
Let $R$ be a commutative Noetherian ring with unit. We classify the characters of the group $\mathrm{EL}_d (R)$ provided that $d$ is greater than the stable range of the ring $R$. It follows that every character of $\mathrm{EL}_d (R)$ is induced from a finite dimensional representation. Towards our main result we classify $\mathrm{EL}_d (R)$-invariant probability measures on the Pontryagin dual group of $R^d$.
Let R be a ring with identity, (M;\leq) a commutative positive strictly ordered monoid and w_m an automorphism for each m \in M . The skew generalized power series ring R[[M,w]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev Neumann Laurent series rings. If S\subset R is a multiplicative set, then R is called right S-Noetherian, if for each ideal I of R, Is \subseteq J\subseteq I for some s\in S and some finitely generated right ideal J . Unifying and generalizing a number of known results, we s…
We define when a noetherian ring R is called a right ( or a left) weakly krull symmetric ring . We then prove that if R is a right ( or a left ) krull homogenous ring then R is a right ( or a left ) weakly krull symmetric ring . This result modifies the main result of [2] . The key terms introduced in this paper are of independent interest .
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 study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up algebras.
R is called a right WV -ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV -ring, then R is right uniform or a right V -ring. It is shown that for a right WV-ring R, R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS or noetherian module. For a finitely generated module M with projective socle over a V -ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X \oplus T, where X is semisimple and T is noetherian with zero socle. I…
It is proved that a ring $R$ is a right uniserial, right Noetherian centrally essential ring if and only if $R$ is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings. Victor Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.
It is proved that a ring $A$ is a right or left Noetherian, right distributive centrally essential ring if and only if $A=A_1\times\cdots\times A_n$, where each of the rings $A_i$ is either a commutative Dedekind domain or a uniserial Artinian centrally essential (not necessarily commutative) ring. V.T.Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. A.A.Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.
We study skew inverse power series extensions R[[y^{-1};tau,delta]], where R is a noetherian ring equipped with an automorphism tau and a tau-derivation delta. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete local, noetherian, Auslander regular domains whose right Krull dimension, global dimension, and classical Krull dimension, are all equal to 2n.
Two constructions are given that describe respectively all shortest primary decompositions and all shortest uniform decompositions for left Noetherian rings. They show that these decompositions are, in general, highly non-unique.
In this note we show that if a noetherian ring R is left and right Krull-homogenous and if: Λ={P\textexclamdown ε spec.R/ |R/P\textexclamdown|_r =|R|_r} and v ={Qj ε spec.R| |R/Qj|l=|R|l} and P =\cap P\textexclamdownε ΛP\textexclamdown and Q = \cap QjεVQj then the following hold true; (1) Pn =0, for some integer n\geq1 (2) Λ=v
A ring R shall be called F-noetherian if every finite subset of R is contained in a (left and right) noetherian subring of R . For example, every commutative ring is tightly F-noetherian in the sense that every finite subset of R generates a noetherian subring of R . F-noetherian rings have many interesting linear algebra properties which we refer to as the full strong rank condition, fully stably finite, and more generally the basic condition. We also study some basic ring-theoretic properties of F-noetherian rings such as localizations of F-noetherian rings. The F-noetherian property is pres…
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 vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ such that two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We characterize commutative Noetherian rings $R$ whose annihilating-ideal graphs have finite genus $γ(\Bbb{AG}(R))$. It is shown that if $R$ is a Noetherian ring such that $0<γ(\Bbb{AG}(R))<\infty$, then $R$ has only finitely many ideals.
Recently, tilting and cotilting classes over commutative noetherian rings have been classified in arXiv:1203.0907. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n-cotilting module inducing C. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which f…
In this paper we prove our main theorem, namely, theorem (8), which states that a link Q\rightarrowP, of prime ideals Q and P of a noetherian ring R that are σ-semistable with respect to a fixed automorphism σ of R, induces a link Q0\rightarrowP0 of the semiprime ideals Q0 and P0 of the ring R,where Q0 and P0 are the largest σ- invariant or σ- stable ideals contained in the prime ideals Q and P. We also prove a converse to this theorem.
In this paper for a noetherian ring R with nilradical N we define semiprime ideals P and Q called as the left and right krull homogenous parts of N . We also recall the known definitions of localisability and the weak ideal invariance (w.i.i for short ) of an ideal of a noetherian ring R . We then state and prove results that culminate in our main theorem whose statement is given below ; Theorem :- Let R be a noetherian ring with nilradical N . Let P and Q be semiprime ideals of R that are the right and left krull homogenous parts of N respectively . Then the following conditions are equivalen…
We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.
The injective spectrum is a topological space associated to a ring $R$, which agrees with the Zariski spectrum when $R$ is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) and establish some basic topological results and a functoriality result, as well as links between the topology and the Krull dimension of the ring (in the sense of Gabriel and Rentschler). Finally, we use these results to compute a number of examples.
We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of $V^*(R/J(R))$, is isomorphic to the monoid of solutions in $(\No \cup\{\infty\})^k$ of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if $M$ is a submonoid of $(\No \cup\{\infty\})^k$ containing an order unit $(n_1,..., n_k)$ of $\No^k$ which is the set of solutions of a system of congruences and linear diop…
We construct a ring with the properties of the title of the paper. We also construct some other local rings of embedding dimension 4 with exotic properties. Among the methods used are the {\tt Macaulay2}-package {\tt DGAlgebras} by Frank Moore, combined with and inspired by results by Anick, Avramov, Backelin, Katthän, Lemaire, Levin, Löfwall and others.
In this paper we study representation theory of the category FI$^m$ introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI$^m$ in a natural way. In particular, we prove the representation stability property of finitely generated FI$^m$-modules over fields of characteristic 0.
We introduce in this work, the class of commutative rings whose lattice of ideals forms an MTL-algebra which is not necessary a BL-algebra. The so-called class of rings will be named MTL-rings. We prove that a local commutative ring with identity is an MTL-ring if and only if it is an arithmetical ring. It is shown that a noetherian commutative ring R with an identity is an MTL-ring if and only if ideals of the localization RM at a maximal ideal M are totally ordered by the set inclusion. Remarking that noetherian MTL-rings are again BL-rings, we work outside of the noetherian case by consider…
Let $i: A\to R$ be a ring morphism, and $χ: R\to A$ a right $R$-linear map with $χ(χ(r)s)=χ(rs)$ and $χ(1_R)=1_A$. If $R$ is a Frobenius $A$-ring, then we can define a trace map $\tr: A\to A^R$. If there exists an element of trace 1 in $A$, then $A$ is right FBN if and only if $A^R$ is right FBN and $A$ is right noetherian. The result can be generalized to the case where $R$ is an $I$-Frobenius $A$-ring, and the condition on the trace can be replaced by a weaker condition. We recover results of Garc\'ıa and del R\'ıo and by Dǎscǎlescu, Kelarev and Torrecillas on actions of group and Hopf algeb…
We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated modules of bounded projective dimension are classified. We also relate our results to Hochster's conjecture on the existence of finitely generated maximal Cohen-Macaulay modules.
We study prime ideals in skew power series rings $T:=R[[y;τ,δ]]$, for suitably conditioned right noetherian complete semilocal rings $R$, automorphisms $τ$ of $R$, and $τ$-derivations $δ$ of $R$. These rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern "Cutting Down" and "Lying Over." In particular, under the additional assumption that $δ= τ- id$ (a basic feature of the Iwasawa-theoretic context), we prove: If $I$ is an ideal of $R$, then there exists a prime ideal $P$ of $S$ contracting to $I$ if and only if $I$ is a $δ$-stable $τ…
Given a complete local Noetherian ring $(A,\m_A)$ with finite residue field and a subfield $\pmb{k}$ of $A/\m_A$, we show that every closed subgroup $G$ of $GL_n(A)$ such that $G\mod{\m_A}\supseteq SL_n(\pmb{k})$ contains a conjugate of $SL_n(W(\pmb{k})_A)$ under some small restrictions on $\pmb{k}$. Here $W(\pmb{k})_A$ is the closed subring of $A$ generated by the Teichmüller lifts of elements of the subfield $\pmb{k}$.
We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings. We characterize noetherian rings via superstability of the class of left modules with embeddings. $\mathbf{Theorem.}$ For a ring $R$ the following are equivalent. - $R$ is left noetherian. - The class of left $R$-modules with embeddings is superstable. - For every $λ\geq |R| + \aleph_0$, there is $χ\geq λ$ such that the class of left $R$-modules with embeddings has uniqueness of limit models of cardinality $χ$. - Every limit model in the class of left $R$-modules…
In a series of papers \cite{BRV1}, \cite{BRV2}, \cite{BRV3} we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "$\bar q$-characteristic coefficient-absorbing polynomial in each T-ideal $Γ$," i.e., a non-identity of the representable algebra $A$ arising from $Γ$, whose ideal of evaluations in $…
Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-modules is a Serre subcategory. In this paper, it is proved that this holds whenever R is a commutative noetherian ring. This paper also yields a module version of the bijection between the set of localizing subcategori…
Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many $\mathbb{N}$-graded rings of finite global dimension; in particular, we see that a noetherian Arti…
Cotorsion pairs and Tor-pairs over commutative noetherian rings
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Centre de Recerca Matemàtica（数学研究中心）；Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）；Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）
This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure s…
It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck's duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of acyclic complexes of projectives by its subcategory of totally acyclic complexes and the corresponding category consisting of injective modules are equivalent. A new characterization is provided for comple…
In this paper we introduce the definition of a noetherian disjoint ring and that of a noetherian non-disjoint ring . For a noetherian ring R , with nilradical N if P and Q represent the semiprime ideals of R called as the right and the left krull-homogenous parts of N as defined in [8] , then we prove the main theorem of this paper for the ring R whose statement is given below. Main Theorem :- Let R be a Noetherian ring with nilradical N . Let P and Q represent the right and the left krull-homogenous parts of N . Then the following hold true for the ring R ; (a) If R is a disjoint ring , then …
Let A be a semprime, right noetherian ring equipped with an automorphism alpha, and let B := A[[y; alpha]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.
We introduce the notion of integrality of Grothendieck categories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of locally noetherian schemes. Two different spaces associated to a Grothendieck category yield respective definitions of integrality, and we prove the equivalence of these definitions using a Grothendieck-categorical version of Gabriel's correspondence, which originally related indecomposable injective modules and prime two-sided ideals for noetherian rings. The generalization of prime two-sided ideals is also used to classi…
Let $R$ be a {\em differentiably simple Noetherian commutative} ring of characteristic $p>0$ (then $(R, \gm)$ is local with $n:= {\rm emdim} (R)<\infty$). A short proof is given of the Theorem of Harper \cite{Harper61} on classification of differentiably simple Noetherian commutative rings in prime characteristic. The main result of the paper is that there {\em exists} a {\em nilpotent simple} derivation of the ring $R$ such that if $\d^{p^i}\neq 0$ then $\d^{p^i}(x_i)=1$ for some $x_i\in \gm$. The derivation $\d $ is given {\em explicitly}, it is {\em unique} up to the action of the group ${\…
A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in [M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra 345 (2011) 257--265] the authors considered this question in the context of finite direct products of commutative Noetherian lo…
