24 results
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2. A Note on the Artinian Cofinite Modules.
- Author
-
Abazari, Nemat and Bahmanpour, Kamal
- Subjects
ARTIN rings ,MODULES (Algebra) ,MATHEMATICAL proofs ,GENERALIZATION ,NOETHERIAN rings ,RING theory ,LOCAL rings (Algebra) - Abstract
In this paper we shall prove the following result, which is a generalization of the Melkersson's main result proved in [16]. Let (R, 𝔪) be a Noetherian local ring such thatis integral overR. LetIbe a proper ideal ofRandAbe an ArtinianR-module. ThenAisI-cofinite if and only if Rad(I + AnnR(A)) = 𝔪. Also, we present an example to show that this result does not hold for an arbitrary local Noetherian ring in general. As an application of this result we prove the following generalization of the Lichtenbaum-Hartshorne Vanishing Theorem (see [5, Theorem 8.2.1]). Let (R, 𝔪) be a Noetherian local ring such thatis integral overR. LetIan ideal ofRandMbe a nonzero finitely generatedR-module of dimensionn. Then the following conditions are equivalent: (i). (ii) There exists a prime ideal 𝔭 in AsshR(M) such that Rad(𝔭 +I) = 𝔪. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
3. Some results on the cofiniteness and annihilators of local cohomology modules.
- Author
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Rastgoo, Fahimeh and Nazari, Alireza
- Subjects
NOETHERIAN rings ,RING theory ,COHEN-Macaulay rings ,COHOMOLOGY theory ,INTEGERS - Abstract
Let (
R ,픪 ) be a Noetherian local ring. In this paper, we give a new characterization for the universal catenaricity ofR and the Cohen-Macaulayness of all its formal fibers, and we study the cofiniteness and annihilators of local cohomology modules. [ABSTRACT FROM AUTHOR]- Published
- 2018
- Full Text
- View/download PDF
4. von Neumann regular modules.
- Author
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Jayaram, C. and Tekir, Ünsal
- Subjects
VON Neumann algebras ,MODULES (Algebra) ,NOETHERIAN rings ,RING theory ,MATHEMATICAL analysis - Abstract
In this paper, we introduce von Neumann regular modules and give many characterizations of von Neumann regular modules. Further, we investigate the relations between von Neumann regular modules and other classical modules. Finally, we characterize Noetherian von Neumann regular modules. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
5. Asymptotic Periodicity of Primes Associated to Multigraded Modules.
- Author
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Hayasaka, Futoshi
- Subjects
MODULES (Algebra) ,RING theory ,NOETHERIAN rings ,MATHEMATICAL proofs ,NUMERICAL analysis ,MATHEMATICAL models - Abstract
In this paper, we investigate the asymptotic behavior of the set of primes associated to a graded ring extension of Noetherian multigraded rings and modules, and prove that the periodicity occurs in a cone. We also prove the same asymptotic behavior of the grade. The previous known results on this subject are recovered as a special case. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
6. C-Pure Projective Modules.
- Author
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Behboodi, M., Ghorbani, A., Moradzadeh-Dehkordi, A., and Shojaee, S.H.
- Subjects
PROJECTIVE modules (Algebra) ,RING theory ,NOETHERIAN rings ,PRINCIPAL ideal domains ,NONCOMMUTATIVE rings ,MATHEMATICAL analysis ,NUMERICAL analysis - Abstract
This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ringRis left Noetherian if and only if every C-pure projective leftR-module is pure projective. Also, over a left hereditary Noetherian ringR, a leftR-moduleMis C-pure projective if and only ifM = N⊕P, whereNis a direct sum of cyclic modules andPis a projective leftR-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that ifRis a local duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis a principal ideal ring. IfRis a left perfect duo-ring, then the C-pure projective leftR-modules and the pure projective leftR-modules coincide if and only ifRis left Köthe (i.e., every leftR-module is a direct sum of cyclic modules). Also, it is shown that for a ringR, if every C-pure projective leftR-module is RD-projective, thenRis left Noetherian, every p-injective leftR-module is injective and every p-flat rightR-module is flat. Finally, it is shown that ifRis a left p.p-ring and every C-pure projective leftR-module is RD-projective, thenRis left Noetherian hereditary. The converse is also true whenRis commutative, but it does not hold in the noncommutative case. [ABSTRACT FROM PUBLISHER]
- Published
- 2013
- Full Text
- View/download PDF
7. On Generalized Cohen–Macaulay Canonical Modules.
- Author
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Hong Loan, NguyenThi and Nhan, LeThanh
- Subjects
GENERALIZATION ,COHEN-Macaulay modules ,NOETHERIAN rings ,RING theory ,QUOTIENT rings ,GORENSTEIN rings ,MATHEMATICAL sequences - Abstract
Let (R, 𝔪) be a Noetherian local ring which is a quotient of a Gorenstein local ring. A finitely generatedR-moduleMis calledgeneralized Cohen–Macaulay canonicalif the canonical moduleK(M) ofMis generalized Cohen–Macaulay. In this paper, we give characterizations of generalized Cohen–Macaulay canonical modules in term of certain systems of parameters. [ABSTRACT FROM PUBLISHER]
- Published
- 2013
- Full Text
- View/download PDF
8. Differential Criterion of Complete Regular Local Rings.
- Author
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Furuya, Mamoru and Niitsuma, Hiroshi
- Subjects
- *
LOCAL rings (Algebra) , *NOETHERIAN rings , *RING theory , *DIFFERENTIAL algebra , *ALGEBRAIC field theory , *ALGEBRA - Abstract
In the paper by Furuya and Niitsuma [Furuya, M., Niitsuma, H. (2002a). On rn-adic higher differentials and regularities of Noetherian complete local rings. J. Math. Kyoto Univ. 42(1):33-40], we gave a regularity criterion of complete Noetherian local rings in terms of the concept of m-adic higher differentials with some assumptions of separability on the residue fields of the local rings. The aim of this paper is to try to weaken "the separability assumptions" in a premise of the theorems in Furuya and Niitsuma (2002a). [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
9. Cohomological Dimension of Complexes#.
- Author
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Dibaei, Mohammad T. and Yassemi, Siamak
- Subjects
- *
HOMOLOGY theory , *MODULES (Algebra) , *NOETHERIAN rings , *COMMUTATIVE rings , *ALGEBRAIC topology , *RING theory - Abstract
In the derived category of the category of modules over a commutative Noetherian ring R, we define, for an ideal [afr] of R, we investigate the interplay between the two naturally defined numbers of cohomological dimensions of a complex X in a certain subcategory of the derived category, namely cd([afr], X) = sup{cd([afr], Hℓ(X)) − ℓ|ℓ ∈ [Zopf]} and− infRΓ[afr](X). Here cd([afr], M) = sup{ℓ ∈ [Zopf]| for an R-module M, and infRΓ[afr](X) denotes the homological infimum of the complex RΓ[afr](X). In this paper, it is shown, among other things, that, for any complex X bounded to the left,− infRΓ[afr](X) ≤ cd([afr], X) and equality holds if indeed H(X) is finitely generated. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
10. Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings#.
- Author
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Asadollahi, Javad and Salarian, Shokrollah
- Subjects
- *
GORENSTEIN rings , *NOETHERIAN rings , *MODULES (Algebra) , *COMMUTATIVE rings , *RING theory , *ALGEBRA - Abstract
Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
11. Homological Characterizations of Rings with Property (P).
- Author
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Huang, Zhaoyong and Tang, Gaohua
- Subjects
- *
COMMUTATIVE rings , *RING theory , *ALGEBRA - Abstract
A commutative ring R is said to satisfy property (P) if every finitely generated proper ideal of R admits a non-zero annihilator. In this paper we give some necessary and sufficient conditions that a ring satisfies property (P). In particular, we characterize coherent rings, noetherian rings and Π-coherent rings with property (P). [ABSTRACT FROM AUTHOR]
- Published
- 2003
- Full Text
- View/download PDF
12. On Nonnil-Noetherian Rings.
- Author
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Badawi, Ayman
- Subjects
- *
NOETHERIAN rings , *RING theory - Abstract
Let R be a commutative ring with 1 such that Nil(R) is a divided prime ideal of R. The purpose of this paper is to introduce a new class of rings that is closely related to the class of Noetherian rings. A ring R is called a Nonnil-Noetherian ring if every nonnil ideal of R is finitely generated. We show that many of the properties of Noetherian rings are also true for Nonnil-Noetherian rings; we use the idealization construction to give examples of Nonnil-Noetherian rings that are not Noetherian rings; we show that for each n ≥ 1, there is a Nonnil-Noetherian ring with Krull dimension n which is not a Noetherian ring. [ABSTRACT FROM AUTHOR]
- Published
- 2003
- Full Text
- View/download PDF
13. HIGHER PRODUCT LEVELS OF NONCOMMUTATIVE RINGS.
- Author
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Cimprič, Jaka
- Subjects
- *
RING theory , *NOETHERIAN rings , *QUADRATIC forms - Abstract
The aim of this paper is to introduce the notion of the nth product level ps[SUBn], of an associative unital ring and study its properties. Our main results are that every Noetherian ring A with ps[SUBn](A) < ∞ for some n ∈ N has ps[SUBnl](A) < ∞ for every odd number l (Theorem 8) and that for every even n ∈ N there exists a skewfield D with ps[SUBn](D) = 1 and ps[SUB2n] (D) = ∞ (Theorem 9). This is in a sharp contrast with the commutative case. Namely, by Proposition 4.6 in (1), for every commutative unital ring R with ps[SUB2](R) < ∞ we have that ps[SUBn](R) < ∞ for every n ∈ N. [ABSTRACT FROM AUTHOR]
- Published
- 2001
- Full Text
- View/download PDF
14. The completion and Krull’s generalized principal ideal theorem on r-Noetherian rings.
- Author
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Chang, Gyu Whan and Kang, Byung Gyun
- Subjects
NOETHERIAN rings ,MATHEMATICS theorems ,RING theory ,COMMUTATIVE rings ,ALGEBRA - Abstract
A ring is called an
r-Noetherian ring if every regular ideal is finitely generated. LetR be an r-Noetherian ring, letI be a regular ideal ofR , and letbe the I -adic completion ofR . We show thatis a Noetherian ring and dim = sup{r-ht( M )∣M ∈Max(R ) andI ⊆M }. LetP be a prime ideal ofR . We also prove that for anya ∈reg (P ), r-htP = ht(P ∕aR )+1 and that ifP is minimal over ann -generated regular ideal, then r-htP ≤n . [ABSTRACT FROM AUTHOR]- Published
- 2018
- Full Text
- View/download PDF
15. Weakly left localizable rings.
- Author
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Bavula, V. V.
- Subjects
RING theory ,MATHEMATICAL analysis ,NOETHERIAN rings ,IDEALS (Algebra) ,MAXIMAL functions - Abstract
A new class of rings,the class of weakly left localizable rings, is introduced. A ringRis calledweakly left localizableif each non-nilpotent element ofRis invertible in some left localizationS−1Rof the ringR. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
16. Hochster’s small MCM conjecture for three-dimensional weakly F-split rings.
- Author
-
Schoutens, Hans
- Subjects
RING theory ,COHEN-Macaulay modules ,NOETHERIAN rings ,LOCAL rings (Algebra) ,FROBENIUS algebras - Abstract
We prove Hochster’s small maximal Cohen–Macaulay conjecture for three-dimensional complete F-pure rings. We deduce this from a more general criterion, and show that only a weakening of the notion of F-purity is needed, to wit, being weakly F-split. We conjecture that any complete ring is weakly F-split. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
17. When Essential Extensions of Finitely Generated Modules are Finitely Generated.
- Author
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Dehghani, N. and Vedadi, M. R.
- Subjects
NOETHERIAN rings ,ASSOCIATIVE rings ,COMMUTATIVE rings ,RING theory ,GORENSTEIN rings - Abstract
For certain classes 𝒞 ofR-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/Mis Noetherian for any finitely generated moduleMR. RingsRwith latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. IfRis a quasi continuous ring in 𝒬 thenR ≃ A ⊕ TwhereAis a semisimple Artinian ring andT ∈ 𝒬 with Z(TT) ≤essTT. [ABSTRACT FROM PUBLISHER]
- Published
- 2016
- Full Text
- View/download PDF
18. Neat-flat Modules.
- Author
-
Büyükaşık, Engin and Durğun, Yılmaz
- Subjects
MODULES (Algebra) ,RING theory ,MORPHISMS (Mathematics) ,KERNEL (Mathematics) ,MATHEMATICAL mappings ,SURJECTIONS ,NOETHERIAN rings - Abstract
LetRbe a ring. A rightR-moduleMis said to be neat-flat if the kernel of any epimorphismY → Mis neat inY, i.e., the induced map Hom(S,Y) → Hom(S,M) is surjective for any simple rightR-moduleS. Neat-flat rightR-modules are projective if and only ifRis a right-CSring. Every cyclic neat-flat rightR-module is projective if and only ifRis rightCSand rightC-ring. It is shown that, over a commutative Noetherian ringR, (1) every neat-flat module is flat if and only if every absolutely coneat module is injective if and only ifR ≅ A × B, whereinAis aQF-ring andBis hereditary, and (2) every neat-flat module is absolutely coneat if and only if every absolutely coneat module is neat-flat if and only ifR ≅ A × B, whereinAis aQF-ring andBis Artinian withJ2(B) = 0. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
19. Some Results on Generalized Local Cohomology Modules.
- Author
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Vahidi, Alireza and Aghapournahr, Moharram
- Subjects
GENERALIZATION ,COHOMOLOGY theory ,MODULES (Algebra) ,COMMUTATIVE rings ,NOETHERIAN rings ,RING theory - Abstract
LetRbe a commutative Noetherian ring with nonzero identity, 𝔞 an ideal ofR,Ma finiteR–module,Xan arbitraryR–module, andna non-negative integer. Here, we show that, in the Serre subcategories of the category ofR–modules, how the generalized local cohomology modules, the ordinary local cohomology modules, and the extension modules behave similarly at the initial pointsi ≤ n. We conclude with some Artinianness and cofiniteness results for, and some finiteness results forand. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
20. Absolutely s -Pure Modules and Neat-Flat Modules.
- Author
-
Büyükaşık, Engin and Durğun, Yılmaz
- Subjects
MODULES (Algebra) ,RING theory ,SUBMODULAR functions ,NOETHERIAN rings ,MATHEMATICAL functions ,MATHEMATICAL models - Abstract
LetRbe a ring with an identity element. We prove thatRis right Kasch if and only if injective hull of every simple rightR-modules is neat-flat if and only if every absolutely pure rightR-module is neat-flat. A commutative ringRis hereditary and noetherian if and only if every absolutelys-pureR-module is injective andRis nonsingular. If every simple rightR-module is finitely presented, then (1)RRis absolutelys-pure if and only ifRis right Kasch and (2)Ris a right-CSring if and only if every pure injective neat-flat rightR-module is projective if and only if every absolutelys-pure leftR-module is injective andRis right perfect. We also study enveloping and covering properties of absolutelys-pure and neat-flat modules. The rings over which every simple module has an injective cover are characterized. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
21. Noncommutative Gröbner Bases over Rings.
- Author
-
Mialebama Bouesso, AndréSaint Eudes and Sow, Djiby
- Subjects
COMMUTATIVE rings ,RING theory ,NOETHERIAN rings ,FUNDAMENTAL theorem of calculus ,ALGORITHMS ,MATHEMATICAL models - Abstract
In this work, we propose a method for computing noncommutative Gröbner bases over a noetherian valuation ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Gröbner bases is generalized to Buchberger's algorithm overR= 𝒱 ⟨ x1,…,xm ⟩, a free associative algebra with noncommuting variables, where 𝒱 = ℤ/nℤ and 𝒱 = ℤ. The proposed process generalizes previous known techniques for the computation of commutative Gröbner bases over a nætherian valuation ring and/or noncommutative Gröbner bases over a field. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
22. Completions of Hypersurface Domains.
- Author
-
Ahn, JiWon, Ferme, E., Jiang, Feiqi, Loepp, S., and Tran, G.
- Subjects
HYPERSURFACES ,MATHEMATICAL domains ,MATHEMATICAL regularization ,NOETHERIAN rings ,RING theory ,MAXIMAL ideals ,INTEGRAL domains - Abstract
LetVbe a complete regular local (Noetherian) ring and letf ∈ Vbe a nonunit. We find necessary and sufficient conditions forto be the completion with respect to the maximal ideal of an integral domain of the formwhereSis a regular local ring whose completion with respect to its maximal ideal isV. In addition, ifcontains the rationals, we give necessary and sufficient conditions forto be the completion of an excellent local domain of the form, wheref ∈ S, andSis a regular local ring whose completion isV. [ABSTRACT FROM PUBLISHER]
- Published
- 2013
- Full Text
- View/download PDF
23. The Noetherian Properties of the Rings of Differential Operators on Central 2-Arrangements.
- Author
-
Nakashima, Norihiro
- Subjects
NOETHERIAN rings ,DIFFERENTIAL operators ,MATHEMATICAL proofs ,HYPERPLANES ,STATISTICAL association ,RING theory ,MATHEMATICAL analysis - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
24. Modules Which are Isomorphic to Their Factor Modules.
- Author
-
Oman, Greg and Salminen, Adam
- Subjects
DEDEKIND rings ,COMMUTATIVE rings ,ARTIN rings ,ASSOCIATIVE rings ,NOETHERIAN rings ,RING theory - Abstract
LetRbe commutative ring with identity and letMbe an infinite unitaryR-module. CallM homomorphically congruent(HC for short) providedM/N ≅ Mfor every submoduleNofMfor which |M/N| = |M|. In this article, we study HC modules over commutative rings. After a fairly comprehensive review of the literature, several natural examples are presented to motivate our study. We then prove some general results on HC modules, including HC module-theoretic characterizations of discrete valuation rings, almost Dedekind domains, and fields. We also provide a characterization of the HC modules over a Dedekind domain, extending Scott's classification over ℤ in [22]. Finally, we close with some open questions. [ABSTRACT FROM PUBLISHER]
- Published
- 2013
- Full Text
- View/download PDF
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