3,149 results on '"Noetherian"'
Search Results
2. An antichain of monomial ideals in a twisted commutative algebra.
- Author
-
Laudone, Robert P.
- Subjects
- *
PARTIALLY ordered sets , *OPEN-ended questions , *COMMUTATIVE algebra , *POLYNOMIAL rings - Abstract
We resolve an open question posed by Nagpal, Sam and Snowden [Selecta. Math. (N.S.) 22 (2016), pp. 913–937] in 2015 concerning a Gröbner theoretic approach to the noetherianity of the twisted commutative algebra Sym(Sym^2(\mathbf {C}^\infty)). We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well-studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Rings of very strong finite type.
- Author
-
Coykendall, Jim and Dutta, Tridib
- Subjects
- *
FINITE rings , *POWER series , *COMMUTATIVE rings - Abstract
The SFT (for strong finite type) condition was introduced by [J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973) 299–304] in the context of studying the condition for formal power series rings to have finite Krull dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. Many others have studied this condition in the context of the dimension of formal power series rings. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub as the VSFT (for very strong finite type) property. As is true of the SFT property, the VSFT property is a property of an ideal that may be extended to a global property of a commutative ring with identity. Any ideal (respectively, ring) that has the VSFT property has the SFT property. In this paper, we explore the interplay of the SFT property and the VSFT property. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Adjacency-like conditions and induced ideal graphs.
- Author
-
Al-Kaseasbeh, Saba and Coykendall, Jim
- Subjects
- *
GRAPH theory , *DATA visualization - Abstract
In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of the associated graph. Behavior of these graphs under standard ring extensions are studied, and in conjunction with the theory, some classical results and connections are made. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. Domains whose ideals meet a universal restriction.
- Author
-
Zafrullah, Muhammad
- Subjects
- *
INTEGRAL domains , *GENERALIZATION - Abstract
Let S (D) represent a set of proper nonzero ideals I (D) (respectively, t-ideals I t (D)) of an integral domain D ≠ q f (D) and let P be a valid property of ideals of D. We say S(D) meets P (denoted S (D) ◃ P) if each s ∈ S (D) is contained in an ideal satisfying P. If S(D) ◃ P , dim (D) cannot be controlled. When R = D [ X ] , I (D) ◃ P does not imply I(R) ◃ P while I t (D) ◃ P implies I t (R) ◃ P usually. We say S(D) meets P with a twist (written S (D) ◃ t P) if each s ∈ S (D) is such that, for some n ∈ N , s n is contained in an ideal satisfying P and study S (D) ◃ t P , as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. Structure and isomorphisms of quantum generalized Heisenberg algebras.
- Author
-
Lopes, Samuel A. and Razavinia, Farrokh
- Subjects
- *
LINEAR algebra , *ALGEBRA , *REPRESENTATIONS of algebras , *AUTOMORPHISM groups - Abstract
In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter q and two polynomials f , g. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. An abstract factorization theorem and some applications.
- Author
-
Tringali, Salvatore
- Subjects
- *
INDECOMPOSABLE modules , *FACTORIZATION , *MONOIDS , *NUMBER theory , *COMMUTATIVE rings , *COMBINATORICS - Abstract
We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular, we obtain a generalization, from cancellative to Dedekind-finite (commutative or non-commutative) monoids, of a classical theorem on "atomic factorizations" that traces back to the work of P.M. Cohn in the 1960s; recover a theorem of D.D. Anderson and S. Valdes-Leon on "irreducible factorizations" in commutative rings; improve on a theorem of A.A. Antoniou and the author that characterizes atomicity in certain "monoids of sets" naturally arising from additive number theory and arithmetic combinatorics; and give a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring is a direct sum of finitely many indecomposable modules (this is in fact a special case of a more general decomposition theorem for the objects of certain categories with finite products). [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
8. Commutative Non-Noetherian Rings with the Diamond Property.
- Author
-
Iovanov, Miodrag C.
- Abstract
A ring R is said to have property (◇) if the injective hull of every simple R-module is locally Artinian. By landmark results of Matlis and Vamos, every commutative Noetherian ring has (◇). We give a systematic study of commutative rings with (◇), We give several general characterizations in terms of co-finite topologies on R and completions of R. We show that they have many properties of Noetherian rings, such as Krull intersection property, and recover several classical results of commutative Noetherian algebra, including some of Matlis and Vamos. Moreover, we show that a complete rings has (◇) if and only if it is Noetherian. We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. Riesz and pre-Riesz monoids.
- Author
-
Zafrullah, Muhammad
- Subjects
- *
MONOIDS , *INTEGRAL domains , *FACTORIZATION - Abstract
Call a directed partially ordered cancellative divisibility monoid M a Riesz monoid if for all x , y 1 , y 2 ≥ 0 in M, x ≤ y 1 + y 2 ⇒ x = x 1 + x 2 where 0 ≤ x i ≤ y i . We explore the necessary and sufficient conditions under which a Riesz monoid M with M + = { x ≥ 0 ∣ x ∈ M } = M generates a Riesz group and indicate some applications. We call a directed p.o. monoid M Π -pre-Riesz if M + = M and for all x 1 , x 2 , ⋯ , x n ∈ M , glb (x 1 , x 2 , ⋯ , x n) = 0 or there is r ∈ Π such that 0 < r ≤ x 1 , x 2 , ⋯ , x n , for some subset Π of M. We explore examples of Π -pre-Riesz monoids of ∗ -ideals of different types. We show for instance that if M is the monoid of nonzero (integral) ideals of a Noetherian domain D and Π the set of invertible ideals, M is Π -pre-Riesz if and only D is a Dedekind domain. We also study factorization in pre-Riesz monoids of a certain type and link it with factorization theory of ideals in an integral domain. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
10. On finiteness properties of Noetherian (Artinian) C*-algebras.
- Author
-
Pourgholamhossein, Mahmood, Rouzbehani, Mohammad, and Amini, Massoud
- Subjects
- *
C*-algebras , *EMBEDDING theorems , *FINITE, The , *ARTIN rings , *DIVISION algebras - Abstract
In this article, we present a trichotomy (a division into three classes) on Noetherian and Artinian C*-algebras and obtain some structural results about Noetherian (and/or Artinian) C*-algebras. We show that every Noetherian, purely infinite and σ-unital C*-algebra A is generated as a C*-ideal by a single projection. We show that if A is a purely infinite, nuclear, separable, Noetherian and Artinian C*-algebra, then A ≅ A ⊗ Z ≅ A ⊗ O ∞ . This is a partial extension of Kirchberg's O ∞ -absorption theorem and Kirchberg's exact embedding theorem. Finally, we show that each Noetherian AF-algebra has a full finite-dimensional C*-subalgebra. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
11. New approaches to finite generation of cohomology rings.
- Author
-
Nguyen, Van C., Wang, Xingting, and Witherspoon, Sarah
- Subjects
- *
HOPF algebras , *FINITE, The , *FINITE fields , *ALGEBRAIC varieties , *COMMUTATIVE algebra , *ALGEBRA - Abstract
In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A , we show that the finite generation condition on A -modules can be replaced by a condition on any affine commutative A -module algebra R under the assumption that R is integral over its invariant subring R A. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A ′ over a finite field, where A can be viewed as a deformation of A ′ , and prove that if the finite generation condition holds for A ′ , then the same condition holds for A. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
12. Noetherian criteria for dimer algebras.
- Author
-
Beil, Charlie
- Subjects
- *
ALGEBRA , *TORUS , *JORDAN algebras - Abstract
Let A be a nondegenerate dimer (or ghor) algebra on a torus, and let Z be its center. Using cyclic contractions, we show the following are equivalent: A is noetherian; Z is noetherian; A is a noncommutative crepant resolution; each arrow of A is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings e i A e i are pairwise isomorphic. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
13. Families of extensions
- Author
-
Emerton, Matthew, author and Gee, Toby, author
- Published
- 2022
- Full Text
- View/download PDF
14. A Note on Analytically Irreducible Domains
- Author
-
Rissner, Roswitha, Fontana, Marco, editor, Frisch, Sophie, editor, Glaz, Sarah, editor, Tartarone, Francesca, editor, and Zanardo, Paolo, editor
- Published
- 2017
- Full Text
- View/download PDF
15. Chain conditions for graph C*-algebras.
- Author
-
Rouzbehani, Mohammad, Pourgholamhossein, Mahmood, and Amini, Massoud
- Subjects
- *
C*-algebras - Abstract
In this article, we study chain conditions for graph C*-algebras. We show that there are infinitely many mutually non isomorphic Noetherian (and Artinian) purely infinite graph C*-algebras with infinitely many ideals. We prove that if E is a graph, then C * (E) {C^{*}(E)} is a Noetherian (resp. Artinian) C*-algebra if and only if E satisfies condition (K) and each ascending (resp. descending) sequence of admissible pairs of E stabilizes. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
16. FI- and OI-modules with varying coefficients.
- Author
-
Nagel, Uwe and Römer, Tim
- Subjects
- *
GROBNER bases , *COMMUTATIVE rings , *POLYNOMIAL rings , *NOETHERIAN rings , *FREE groups , *ALGEBRA , *POLYNOMIALS , *UNIFORMITY - Abstract
We introduce FI-algebras over a commutative ring K and the category of FI-modules over an FI-algebra. Such a module may be considered as a family of invariant modules over compatible varying K -algebras. FI-modules over K correspond to the well studied constant coefficient case where every algebra equals K. We show that a finitely generated FI-module over a noetherian polynomial FI-algebra is a noetherian module. This is established by introducing OI-modules. We prove that every submodule of a finitely generated free OI-module over a noetherian polynomial OI-algebra has a finite Gröbner basis. Applying our noetherianity results to a family of free resolutions, finite generation translates into stabilization of syzygies in any fixed homological degree. In particular, in the graded case this gives uniformity results on degrees of minimal syzygies. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
17. The absolute Euler product representation of the absolute zeta function for a torsion free Noetherian F1-scheme
- Author
-
Takuki Tomita
- Subjects
Noetherian ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Number Theory ,Infinite product ,Function (mathematics) ,Riemann zeta function ,symbols.namesake ,Scheme (mathematics) ,symbols ,Torsion (algebra) ,Limit (mathematics) ,Euler product ,Mathematics - Abstract
The absolute zeta function for a scheme X of finite type over Z satisfying a certain condition is defined as the limit as p → 1 of the zeta function of X ⊗ F p . In 2016, after calculating absolute zeta functions for a few specific schemes, Kurokawa suggested that an absolute zeta function for a general scheme of finite type over Z should have an infinite product structure which he called the absolute Euler product. In this article, formulating his suggestion using a torsion free Noetherian F 1 -scheme defined by Connes and Consani, we give a proof of his suggestion. Moreover, we show that each factor of the absolute Euler product is derived from the counting function of the F 1 -scheme.
- Published
- 2022
18. Choice principles and lift lemmas
- Author
-
Marcel Ern'e
- Subjects
choice ,(super)compact ,foot ,free semilattice ,locale ,noetherian ,prime ,sober ,well-filtered ,Mathematics ,QA1-939 - Abstract
We show that in {bf ZF} set theory without choice, the Ultrafilter mbox{Principle} ({bf UP}) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasi-continuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from {bf UP} but also from {bf DC}, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ${bf ZF}$ set theory.
- Published
- 2017
19. When weak-injective modules decompose like injectives
- Author
-
Fuchs, László
- Published
- 2021
- Full Text
- View/download PDF
20. ON THE QUADRATIC DUAL OF THE FOMIN-KIRILLOV ALGEBRAS.
- Author
-
WALTON, CHELSEA and ZHANG, JAMES J.
- Subjects
- *
ALGEBRA , *NOETHERIAN rings , *POLYNOMIALS , *LOGICAL prediction , *MATHEMATICAL equivalence - Abstract
We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) εn! of the Fomin-Kirillov algebras εn; these algebras are connected N-graded and are defined for n ≥ 2. We establish that the algebra εn! is module finite over its cεnter (and thus satisfies a polynomial idεntity), is Noetherian, and has Gelfand-Kirillov dimεnsion ⌊n/2⌋ for each n ≥ 2. We also observe that εn! is not prime for n ≥ 3. By a result of Roos, εn is not Koszul for n ≥ 3, so neither is εn! for n ≥ 3. Nevertheless, we prove that εn! is Artin-Schelter (AS-)regular if and only if n = 2, and that εn! is both AS-Gorεnstein and AS-Cohεn-Macaulay if and only if n = 2, 3. We also show that the depth of εn! is ≤ 1 for each n ≥ 2, conjecture that we have equality, and show that this claim holds for n = 2, 3. Several other directions for further examination of εn! are suggested at the εnd of this article. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
21. A generalized Noetherian condition for Lie algebras.
- Author
-
Aldosray, Falih A. M. and Stewart, Ian
- Subjects
- *
LIE algebras , *NOETHERIAN rings - Abstract
A Lie algebra (over any field and of any dimension) is Noetherian if it satisfies the maximal condition on ideals. We introduce a new and more general class of quasi-Noetherian Lie algebras that possess several of the main properties of Noetherian Lie algebras. This class is shown to be closed under quotients and extensions. We obtain conditions under which a quasi-Noetherian Lie algebra is Noetherian. Next, we consider various questions about locally nilpotent and soluble radicals of quasi-Noetherian Lie algebras. We show that there exists a semisimple quasi-Noetherian Lie algebra that is not Noetherian. Finally, we consider some analogous results for groups and prove that a quasi-Noetherian group is countably recognizable. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
22. Two extensions of right G-semilocal and right N-semilocal rings.
- Author
-
Fahmy, M. H., Hassanein, A. M., and Abdelwahab, S. M.
- Subjects
- *
JACOBSON radical , *POLYNOMIALS , *ARTIN rings , *LOCAL rings (Algebra) , *NOETHERIAN rings - Abstract
The paper studies the transfer of the property of being right G-semilocal and right N-semilocal from a ring R to its extensions and vice versa. We will focus on two extensions, namely finite normalizing extension S of R and the ring of polynomials R [ x ]. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
23. An integral theory of dominant dimension of Noetherian algebras
- Author
-
Tiago Cruz
- Subjects
Noetherian ,Noetherian ring ,Pure mathematics ,Ring (mathematics) ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Dimension (graph theory) ,Homological algebra ,Invariant (mathematics) ,Schur algebra ,Commutative property ,Mathematics - Abstract
Dominant dimension is introduced into integral representation theory, extending the classical theory of dominant dimension of Artinian algebras to projective Noetherian algebras (that is, algebras which are finitely generated projective as modules over a commutative Noetherian ring). This new homological invariant is based on relative homological algebra introduced by Hochschild in the 1950s. Amongst the properties established here are a relative version of the Morita-Tachikawa correspondence and a relative version of Mueller's characterization of dominant dimension. The behaviour of relative dominant dimension of projective Noetherian algebras under change of ground ring is clarified and we explain how to use this property to determine the relative dominant dimension of projective Noetherian algebras. In particular, we determine the relative dominant dimension of Schur algebras and quantized Schur algebras.
- Published
- 2022
24. Support for Integrable Hopf Algebras via Noncommutative Hypersurfaces
- Author
-
Cris Negron and Julia Pevtsova
- Subjects
Noetherian ,Pure mathematics ,Finite group ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,Hopf algebra ,01 natural sciences ,Noncommutative geometry ,Global dimension ,Tensor product ,Hypersurface ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics - Abstract
We consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated to functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$., 52 pages, minor changes to text
- Published
- 2021
25. Hilbert-Kirby Polynomials in Generalized Local Cohomology Modules
- Author
-
M. Shafiei, A. Azari, Ahmad Khojali, and Naser Zamani
- Subjects
Combinatorics ,Noetherian ,Base (group theory) ,Physics ,Ring (mathematics) ,Mathematics::Commutative Algebra ,Degree (graph theory) ,Primary ideal ,General Mathematics ,Ideal (ring theory) ,Local cohomology ,Quotient - Abstract
Let $R = \oplus _{n\in \mathbb {N}_{0}}R_{n}$ be a Noetherian homogeneous ring with irrelevant ideal $R_{+} = \oplus _{n\in \mathbb {N}} R_{n}$ and with local base ring $(R_{0},\mathfrak {m}_{0})$ . Let M, N be two finitely generated $\mathbb {Z}$ -graded R-modules. We show that the lengths of the graded components of various graded submodules and quotients of the i-th generalized local cohomology $H^{i}_{R_{+}}(M, N)$ are anti-polynomial. Under some mild assumptions, the Artinianness of $H^{i}_{R_{+}}(M, N)$ and the asymptotic behavior of the R0-modules $H^{i}_{R_{+}}(M, N)_{n}$ for $n\rightarrow -\infty $ in the range $i\leq \inf \{i\in \mathbb {N}_{0} \vert \sharp \{n\vert \ell _{R_{0}}$ $(H^{i}_{ R_{+}}(M , N)_{n}) = \infty \}=\infty \}$ will be studied. Moreover, it has been proved that, if u is the least integer i for which $H^{i}_{R_{+}}(M,N)$ is not Artinian and $\mathfrak {q}_{0}$ is an $\mathfrak {m}_{0}$ -primary ideal of R0, then $H^{u}_{R_{+}}(M,N)/\mathfrak q_{0}H^{u}_{R_{+}}(M,$ N) is Artinian with Hilbert-Kirby polynomial of degree less than u. In particular, with M = R, we deduce the correspondent result for ordinary local cohomology module $H^{i}_{R_{+}}(N)$ .
- Published
- 2021
26. Nonnil–Laskerian rings
- Author
-
Samir Moulahi
- Subjects
Power series ,Noetherian ,Class (set theory) ,Ring (mathematics) ,Polynomial ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Geometry and Topology ,Ideal (ring theory) ,Commutative ring ,Algebraic geometry ,Mathematics - Abstract
Let R be a commutative ring with unity. In this paper we introduce the concept of Nonnil–Laskerian ring that is related to the class of Laskerian rings. A ring R is said to be Nonnil–Laskerian if every nonnil ideal I of R is decomposable. We show that Nonnil–Laskerian rings enjoy analogs of many properties of Laskerian ring. We give an example of Nonnil–Laskerian ring, wich is not Laskerian. We study the Nonnil–Laskerian property over the polynomial and formel power series rings. In particular, we show that we have not an equivalence between Nonnil–Laskerian and Nonnil–Noetherian concepts in R[[X]] and R[X], contrary to the Laskerian and Noetherian concepts.
- Published
- 2021
27. Prime thick subcategories and spectra of derived and singularity categories of noetherian schemes
- Author
-
Hiroki Matsui
- Subjects
Noetherian ,Pure mathematics ,Triangulated category ,General Mathematics ,Mathematics - Category Theory ,Algebraic geometry ,Topological space ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Spectrum (topology) ,Prime (order theory) ,Mathematics - Algebraic Geometry ,Singularity ,Mathematics::Category Theory ,FOS: Mathematics ,Noetherian scheme ,13D09, 13H10, 14J60, 18E30 ,Category Theory (math.CT) ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory ,Mathematics - Abstract
For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define the spectrum of $\mathcal{T}$, which shares the basic properties with the spectrum of a tensor triangulated category introduced by Balmer. We mainly focus on triangulated categories that appear in algebraic geometry such as the derived and the singularity categories of a noetherian scheme $X$. We prove that certain classes of thick subcategories are prime thick subcategories of these triangulated categories. Furthermore, we use this result to show that certain subspaces of $X$ are embedded into their spectra as topological spaces., 20 pages
- Published
- 2021
28. SOME STABLE NON-ELEMENTARY CLASSES OF MODULES
- Author
-
Marcos Mazari-Armida
- Subjects
Noetherian ,Class (set theory) ,Ring (mathematics) ,Logic ,Mathematics - Logic ,Mathematics - Rings and Algebras ,Elementary class ,Injective function ,Combinatorics ,Philosophy ,Pure submodule ,Primary: 03C48 Secondary: 03C45, 03C60, 13L05, 16D10, 16P40 ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Torsion (algebra) ,Dedekind cut ,Logic (math.LO) ,Mathematics - Abstract
Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if $T$ is a complete first-order theory extending the theory of modules, then the class of models of $T$ with pure embeddings is stable. In [Maz4, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq_p)$ such that $K$ is a class of modules and $\leq_p$ is the pure submodule relation. In this paper we give some instances where this is true: $\textbf{Theorem.}$ Assume $R$ is an associative ring with unity. Let $(K, \leq_p)$ be an AEC such that $K \subseteq R\text{-Mod}$ and $K$ is closed under finite direct sums, then: - If $K$ is closed under pure-injective envelopes, then $(K, \leq_p)$ is $\lambda$-stable for every $\lambda \geq LS(K)$ such that $\lambda^{|R| + \aleph_0}= \lambda$. - If $K$ is closed under pure submodules and pure epimorphic images, then $(K, \leq_p)$ is $\lambda$-stable for every $\lambda$ such that $\lambda^{|R| + \aleph_0}= \lambda$. - Assume $R$ is Von Neumann regular. If $K$ is closed under submodules and has arbitrarily large models, then $(K, \leq_p)$ is $\lambda$-stable for every $\lambda$ such that $\lambda^{|R| + \aleph_0}= \lambda$. As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy. Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of injective torsion modules., Comment: 22 pages
- Published
- 2021
29. Algebraic Geometry over Algebraic Structures. VIII. Geometric Equivalences and Special Classes of Algebraic Structures
- Author
-
E. Yu. Daniyarova, Alexei Myasnikov, and V. N. Remeslennikov
- Subjects
Statistics and Probability ,Noetherian ,Pure mathematics ,Class (set theory) ,Series (mathematics) ,Algebraic structure ,Applied Mathematics ,General Mathematics ,Algebraic geometry ,Invariant (mathematics) ,Mathematics - Abstract
This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometric, universal geometric, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, qω-compact, uω-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class K, which do not? (2) With respect to which equivalences a given class K is invariant, with respect to which it is not?
- Published
- 2021
30. Relative singularity categories and singular equivalences
- Author
-
Rasool Hafezi
- Subjects
Noetherian ,Subcategory ,Path (topology) ,Noetherian ring ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Homotopy category ,Triangular matrix ,Lift (mathematics) ,Combinatorics ,Singularity ,Mathematics::Category Theory ,Geometry and Topology ,Mathematics - Abstract
Let R be a right noetherian ring. We introduce the concept of relative singularity category $$\Delta _{\mathcal {X} }(R)$$ of R with respect to a contravariantly finite subcategory $$\mathcal {X} $$ of $${\text {{mod{-}}}}R.$$ Along with some finiteness conditions on $$\mathcal {X} $$ , we prove that $$\Delta _{\mathcal {X} }(R)$$ is triangle equivalent to a subcategory of the homotopy category $$\mathbb {K} _\mathrm{{ac}}(\mathcal {X} )$$ of exact complexes over $$\mathcal {X} $$ . As an application, a new description of the classical singularity category $$\mathbb {D} _\mathrm{{sg}}(R)$$ is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.
- Published
- 2021
31. Retractable and coretractable modules in Wisbauer category
- Author
-
M. S. Eryashkin and A. N. Abyzov
- Subjects
Combinatorics ,Noetherian ,Physics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Category Theory ,Coalgebra ,Mathematics::Rings and Algebras ,Sigma ,Field (mathematics) ,Geometry and Topology ,Algebraic geometry ,Algebra over a field - Abstract
A right R-module M is called: (1) retractable if $${{\,\mathrm{Hom}\,}}_R(M, N) \ne 0$$ for any non-zero submodule N of M; (2) coretractable if $${{\,\mathrm{Hom}\,}}_R(M/N, M)\ne 0$$ for any proper submodule N of M. It shows that if M is locally noetherian and every nonzero module in the category $$\sigma [M]$$ has a maximal submodule, then the retractability and coretractability of modules in $$\sigma [M]$$ coincide. Let C be a coalgebra over a field k. We prove that all right C-comodules are retractable if and only if every right C-comodule is coretractable.
- Published
- 2021
32. On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras
- Author
-
José A. Vélez-Marulanda, Hernán Giraldo, Adriana Fonce-Camacho, and Pedro Rizzo
- Subjects
Noetherian ,Derived category ,Deformation ring ,Mathematics::Commutative Algebra ,General Mathematics ,Dimension (graph theory) ,Lambda ,Coherent sheaf ,Combinatorics ,High Energy Physics::Theory ,Residue field ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Algebraically closed field ,Mathematics - Representation Theory ,Mathematics - Abstract
Let Λ be a basic finite dimensional algebra over an algebraically closed field $\Bbbk $ , and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. In this article, we prove that if $\widehat {V}$ is a left $\widehat {\Lambda }$ -module with finite dimension over $\Bbbk $ , then $\widehat {V}$ has a well-defined versal deformation ring $R(\widehat {\Lambda },\widehat {V})$ , which is a local complete Noetherian commutative $\Bbbk $ -algebra whose residue field is also isomorphic to $\Bbbk $ . We also prove that $R(\widehat {\Lambda }, \widehat {V})$ is universal provided that $\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk $ and that in this situation, $R(\widehat {\Lambda }, \widehat {V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $\mathbb {P}^{1}_{\Bbbk }$ .
- Published
- 2021
33. Derived decompositions of abelian categories I
- Author
-
Changchang Xi and Hongxing Chen
- Subjects
Noetherian ,Pure mathematics ,Noetherian ring ,Ring (mathematics) ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,Bounded function ,FOS: Mathematics ,Torsion (algebra) ,Category Theory (math.CT) ,Krull dimension ,Representation Theory (math.RT) ,Primary 16G10, 18E10, 13E05, Secondary 18E30, 13E05 ,Abelian group ,Commutative property ,Mathematics - Representation Theory ,Mathematics - Abstract
Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of abelian categories. We give a sufficient condition for arbitrary abelian categories to have such derived decompositions and show that it is also necessary for abelian categories with enough projectives and injectives. For bounded derived categories, we describe which semi-orthogonal decompositions are determined by derived decompositions. The necessary and sufficient condition is then applied to the module categories of rings: localizing subcategories, homological ring epimorphisms, commutative noetherian rings and nonsingular rings. Moreover, for a commutative noetherian ring of Krull dimension at most $1$, a derived stratification of its module category is established., 27 pages. This is a revision of arXiv:1804.10759. The main results are extended
- Published
- 2021
34. Nakayama automorphisms of graded Ore extensions of Koszul Artin-Schelter regular algebras
- Author
-
Y. Guo and Y. Shen
- Subjects
Noetherian ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,010102 general mathematics ,Superpotential ,Dimension (graph theory) ,Ore extension ,Regular algebra ,Quotient algebra ,Mathematics - Rings and Algebras ,Automorphism ,01 natural sciences ,Mathematics::Group Theory ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
Let $A$ be a Koszul Artin-Schelter regular algebra, $\sigma$ a graded automorphism of $A$ and $\delta$ a degree-one $\sigma$-derivation of $A$. We introduce an invariant for $\delta$ called the $\sigma$-divergence of $\delta$. We describe the Nakayama automorphism of the graded Ore extension $B=A[z;\sigma,\delta]$ explicitly using the $\sigma$-divergence of $\delta$, and construct a twisted superpotential $\hat{\omega}$ for $B$ so that it is a derivation quotient algebra defined by $\hat{\omega}$. We also determine all graded Ore extensions of noetherian Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms.
- Published
- 2021
35. Autonomous Noetherian Boundary-Value Problem in the Case of Parametric Resonance
- Author
-
S. M. Chuiko, O.V. Nesmelova, and O. S. Chuiko
- Subjects
Statistics and Probability ,Noetherian ,Nonlinear system ,Iterative method ,Applied Mathematics ,General Mathematics ,Scheme (mathematics) ,Ordinary differential equation ,Applied mathematics ,Boundary value problem ,Parametric oscillator ,Constructive ,Mathematics - Abstract
We establish constructive conditions for the solvability of a nonlinear autonomous boundary-value problem in the presence of parametric resonance and develop a scheme for the construction of solutions of this problem. We propose a convergent iterative algorithm for finding approximate solutions of the nonlinear autonomous Noetherian boundary-value problem for a system of ordinary differential equations in the case of parametric resonance. As an example of application of the constructed iterative algorithm, we determine approximate solutions of the periodic boundary-value problem for the autonomous Duffing-type equation with parametric perturbation.
- Published
- 2021
36. Intersections of resolving subcategories and intersections of thick subcategories
- Author
-
Ryo Takahashi
- Subjects
Noetherian ,Ring (mathematics) ,Pure mathematics ,Mathematics::Commutative Algebra ,Mathematics::Category Theory ,General Mathematics ,Complete intersection ,Local ring ,Algebraic geometry ,Type (model theory) ,Abelian group ,Commutative property ,Mathematics - Abstract
Let R be a commutative Noetherian local ring. We consider how nontrivial resolving/thick subcategories of abelian/triangulated categories associated to R intersect. It is understood well when R is a complete intersection or a Cohen–Macaulay ring of finite representation type.
- Published
- 2021
37. When a (dual-)Baer module is a direct sum of (co-)prime modules
- Author
-
M. R. Vedadi and Najma Ghaedan
- Subjects
Noetherian ,Ring (mathematics) ,Endomorphism ,Coprime integers ,Direct sum ,General Mathematics ,Mathematics::Rings and Algebras ,Dimension (graph theory) ,Mathematics::General Topology ,Prime (order theory) ,Combinatorics ,Mathematics::Logic ,Mathematics::Group Theory ,Commutative property ,Mathematics - Abstract
Since 2004, Baer modules have been considered by many authors as a generalization of the Baer rings. A module $M_R$ is called Baer if every intersection of the kernels of endomorphisms on $M_R$ is a direct summand of $M_R$. It is known that commutative Baer rings are reduced. We prove that if a Baer module M is a direct sum of prime modules, then every direct summand of M is retractable. The converse is true whenever the triangulating dimension of $M$ is finite (e.g. if the uniform dimension of M is finite). Dually, if every direct summand of a dual-Baer module M is co-retractable, then it is a direct sum of co-prime modules and the converse is true whenever the sum is finite or M is a max-module. Among other applications, we show that if R is a commutative hereditary Noetherian ring then a finitely generated R-module is Baer iff it is projective or semisimple. Also, over a ring Morita equivalent to a perfect duo ring, all dual-Baer modules are semisimple.
- Published
- 2021
38. Local Cohomology Modules and their Properties
- Author
-
Masoumeh Hasanzad and Jafar A'zami
- Subjects
Noetherian ,Functor ,Mathematics::Commutative Algebra ,Cofiniteness ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,0211 other engineering and technologies ,Local ring ,021107 urban & regional planning ,02 engineering and technology ,Local cohomology ,01 natural sciences ,Combinatorics ,Ideal (ring theory) ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
Let (R,m) be a complete Noetherian local ring and let M be a generalized Cohen–Macaulay R-module of dimension d ≥ 2. We show that $$ D\left({H}_m^d\left(D\left({H}_m^d\left({D}_m(M)\right)\right)\right)\right)\approx {D}_m(M), $$ where D = Hom(−,E) and Dm(−) is the ideal transform functor. In addition, by assuming that I is a proper ideal of a local ring R, we obtain some results on finiteness of the Bass numbers, cofiniteness, and cominimaxness of the local cohomology modules with respect to I.
- Published
- 2021
39. The Class of Noetherian RingsWith Finite Valuation Dimension
- Author
-
Hanni Garminia, Pudji Astuti, and Samsul Arifin
- Subjects
Statistics and Probability ,Noetherian ,Computer Science::Computer Science and Game Theory ,Economics and Econometrics ,Pure mathematics ,Noetherian ring ,Ring (mathematics) ,Mathematics::Commutative Algebra ,Mathematics::Rings and Algebras ,Local ring ,Artinian ring ,Valuation (logic) ,Perfect ring ,Dimension (vector space) ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Not a long time ago, Ghorbani and Nazemian [2015] introduced the concept of dimension of valuation which measures how much does the ring differ from the valuation. They've shown that every Artinian ring has a finite valuation dimensions. Further, any comutative ring with a finite valuation dimension is semiperfect. However, there is a semiperfect ring which has an infinite valuation dimension. With those facts, it is of interest to further investigate property of rings that has a finite dimension of valuation. In this article we define conditions that a Noetherian ring requires and suffices to have a finite valuation dimension. In particular we prove that, if and only if it is Artinian or valuation, a Noetherian ring has its finite valuation dimension. In view of the fact that a ring needs a semi perfect dimension in terms of valuation, our investigation is confined on semiperfect Noetherian rings. Furthermore, as a finite product of local rings is a semi perfect ring, the inquiry into our outcome is divided into two cases, the case of the examined ring being local and the case where the investigated ring is a product of at least two local rings. This is, first of all, that every local Noetherian ring possesses a finite valuation dimension, if and only if it is Artinian or valuation. Secondly, any Notherian Ring generated by two or more local rings is shown to have a finite valuation dimension, if and only if it is an Artinian.
- Published
- 2021
40. Finitely Generated Modules's Uniserial Dimensions Over a Discrete Valuation Domain
- Author
-
Pudji Astuti, Samsul Arifin, and Hanni Garminia
- Subjects
Statistics and Probability ,Noetherian ,Economics and Econometrics ,Ring (mathematics) ,Pure mathematics ,Mathematics::Commutative Algebra ,Direct sum ,Mathematics::Rings and Algebras ,Free module ,Commutative ring ,Mathematics::Category Theory ,Domain (ring theory) ,Torsion (algebra) ,Elementary divisors ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
We present some methods for calculating the module's uniserial dimension that finitely generated over a DVD in this article. The idea of a module's uniserial dimension over a commutative ring, which defines how far the module deviates from being uniserial, was recently proposed by Nazemian etc. They show that if R is Noetherian commutative ring, which implies that every finitely generated module over R has uniserial dimension. Ghorbani and Nazemians have shown that R is Noetherian (resp. Artinian) ring if only the ring R X R has (resp. finite) valuation dimension. The finitely generated modules over valuation domain are further examined from here. However, since the region remains too broad, further research into the module's uniserial dimensions that finitely generated over a DVD is needed. In the case of a DVD R, a finitely generated module over R can, as is well-known, be divided into a direct sum of torsion and a free module. Therefore, first, we present methods for determining the primary module's uniserial dimension, and then followed by methods for the general finitely generated module. As can be observed, the module's uniserial dimension is a function of the elementary divisors and the rank of the non torsion module item, which is the major finding of this work.
- Published
- 2021
41. Additive Rank Functions and Chain Conditions
- Author
-
Smith, Patrick F., Albu, Toma, editor, Birkenmeier, Gary F., editor, Erdoğgan, Ali, editor, and Tercan, Adnan, editor
- Published
- 2010
- Full Text
- View/download PDF
42. Prime Ideals in Noetherian Rings: A Survey
- Author
-
Wiegand, Roger, Wiegand, Sylvia, Albu, Toma, editor, Birkenmeier, Gary F., editor, Erdoğgan, Ali, editor, and Tercan, Adnan, editor
- Published
- 2010
- Full Text
- View/download PDF
43. On Primary and Classical Primary Submodules.
- Author
-
Naderi, Mohammad Hasan and Jahani-Nezhad, Reza
- Subjects
- *
MODULES (Algebra) , *NOETHERIAN rings , *COMMUTATIVE rings , *MATHEMATICS , *NUMBER theory - Abstract
Let R be a commutative ring with identity and M be an R-module. In this paper some properties of primary and classical primary submodules will be investigated. It is shown that if F is a faithfully flat R-module, then Q is a primary submodule of M if and only if F ⊗ Q is a primary submodule of F ⊗ M. Also some characterization of classical primary submodules are given. Furthermore, the existence of (minimal) primary submodules containing classical primary submodules are investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2018
44. On right G-semilocal rings.
- Author
-
Fahmy, M. H., Hassanein, A. M., and Abdelwahab, S. M.
- Subjects
SEMILOCAL rings ,FINITE rings ,NOETHERIAN rings ,RING theory ,ALGEBRA - Abstract
The notion of semilocal ring is extended to the classes of right G-semilocal and right N-semilocal rings. We explore the algebraic properties of such classes and study their relations with many other rings such as clean, exchange and I-finite rings. Localization of right G-semilocal rings is considered. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
45. The Cohen Macaulay Property for Noncommutative Rings.
- Author
-
Brown, Ken and Macleod, Marjory
- Abstract
Let R be a noetherian ring which is a finite module over its centre Z( R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z( R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
46. Lifting of recollements and gluing of partial silting sets
- Author
-
Alexandra Zvonareva and Manuel Saorín
- Subjects
Noetherian ,Pure mathematics ,General Mathematics ,01 natural sciences ,Lift (mathematics) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Category Theory (math.CT) ,16E35, 18E30 ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Equivalence (measure theory) ,Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Coproduct ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Rings and Algebras (math.RA) ,Bounded function ,Torsion (algebra) ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.
- Published
- 2021
47. When weak-injective modules decompose like injectives
- Author
-
László Fuchs
- Subjects
Noetherian ,Cyclic module ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Prime ideal ,010102 general mathematics ,Commutative ring ,01 natural sciences ,Weak dimension ,Injective function ,0101 mathematics ,Indecomposable module ,Commutative property ,Mathematics - Abstract
The aim of this note is to find those commutative rings over which an exact analogue of the structure theory of injective modules over commutative noetherian rings holds for weak-injective modules, i.e. for modules M satisfying $$\mathop {\mathrm{Ext}}\nolimits _R^1(A,M)=0$$ for all modules A of weak dimension $$\le 1$$ . We will show that, surprisingly, but a very few commutative rings R possess the property that their weak-injective modules admit (up to isomorphism) unique decompositions into direct sums of indecomposable modules each of which is the injective or the weak-injective envelope of a cyclic module of the form $$R/{\mathsf {p}} $$ with a prime ideal $${\mathsf {p}} $$ .
- Published
- 2021
48. Localization of Hopfian and Cohopfian Objects in the Categories of A − Mod, AGr(A − Mod) and COMP(AGr(A − Mod))
- Author
-
Seydina Ababacar Balde, Mohamed Ben Faraj Ben Maaouia, and Ahmed Ould Chbih
- Subjects
Statistics and Probability ,Noetherian ,Numerical Analysis ,Sequence ,Algebra and Number Theory ,Applied Mathematics ,Multiplicative function ,Graded ring ,Epimorphism ,Theoretical Computer Science ,Combinatorics ,Morphism ,Geometry and Topology ,Isomorphism ,Invariant (mathematics) ,Mathematics - Abstract
The aim of this paper is to study the localization of hopfian and cohopfian objects in the categories A-Mod of left A-modules, AGr(A-Mod) of graded left A-modules and COMP(AGr(A-Mod)) of complex sequences associated to graded left A-modules.We have among others the main following results :1. Let M be a noetherian graded left A-module, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions, N a submodule of M, M_{*} is a noetherian quasi-injective complex sequence associated with M and N_{*} is an essential and completely invariant complex sub\--sequence of M_{*}. Then, S^{-1}(N_{*}) the complex sequence of morphisms of left S^{-1}A\--modules is cohopfian if, and only, if S^{-1}(M_{*}) is cohopfian ;2. let M be a graded left A\--module and S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. If M_{*} is a hopfian, noetherian and quasi-injective complex sequence associated with M, then the complex sequence of morphisms of left S^{-1}(A)-modules S^{-1}(M_{*}) has the following property :{any epimorphism of sub-complex S^{-1}(N_{*}) of S^{-1}(M_{*}) is an isomorphism } ;3. let M be a graded left A-module, N a graded submodule of M, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. M_{*} the quasi-projective complex sequence associated with M and $N_{*}$ a superfluous and completely invariant complex sub\--sequence of $M_{*}$. Then the complex morphism sequence of left $S^{-1}(A)$\--modules $S^{-1}(N_{*})$ is hopfian if, and only if, $S^{-1}(M_{*}/N_{*})$ the complex sequence associated with S^{-1}(M/N) is hopfian.Â
- Published
- 2021
49. Commutative Non-Noetherian Rings with the Diamond Property
- Author
-
Miodrag Cristian Iovanov
- Subjects
Noetherian ,Pure mathematics ,Ring (mathematics) ,Noetherian ring ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,0211 other engineering and technologies ,Local ring ,021107 urban & regional planning ,02 engineering and technology ,Commutative ring ,01 natural sciences ,Simple (abstract algebra) ,Injective hull ,0101 mathematics ,Commutative property ,Mathematics - Abstract
A ring R is said to have property (◇) if the injective hull of every simple R-module is locally Artinian. By landmark results of Matlis and Vamos, every commutative Noetherian ring has (◇). We give a systematic study of commutative rings with (◇), We give several general characterizations in terms of co-finite topologies on R and completions of R. We show that they have many properties of Noetherian rings, such as Krull intersection property, and recover several classical results of commutative Noetherian algebra, including some of Matlis and Vamos. Moreover, we show that a complete rings has (◇) if and only if it is Noetherian. We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples.
- Published
- 2021
50. Cohomological supports over derived complete intersections and local rings
- Author
-
Josh Pollitz
- Subjects
Noetherian ,Derived category ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Polynomial ring ,010102 general mathematics ,Complete intersection ,Local ring ,Koszul complex ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,16. Peace & justice ,Complete intersection ring ,01 natural sciences ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,13D09, 13D07 (primary), 14M10, 18G15 (secondary) ,0101 mathematics ,Exterior algebra ,Mathematics - Abstract
A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples and provide an application by answering a question of D. Jorgensen in the negative., 40 pages. V2: New introduction and minor changes. To appear in Math. Zeit
- Published
- 2021
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.