共查询到20条相似文献,搜索用时 15 毫秒
1.
Amir Mafi 《Proceedings Mathematical Sciences》2009,119(2):159-164
Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H
a
i
(N) is a weakly Laskerian R-module for all i < t, then Hom
R
(R/a, H
a
t
(M, N)) is weakly Laskerian R-module. Also, we prove that Ext
R
i
(R/a, H
a
t
)) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp
R
(H
a
i
(N)) is a finite set for all i < t, then Ext
R
i
(R/a, H
a
t
(N)) is weakly Laskerian R-module for all i = 0, 1. 相似文献
2.
Let R be a noetherian ring,
\mathfraka{\mathfrak{a}} an ideal of R, and M an R-module. We prove that for a finite module M, if
Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is minimax for all i ≥ r ≥ 1, then
Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If
Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is coatomic for i ≤ r (M finite) then
Hi\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing
theorems are proved for local cohomology modules. 相似文献
3.
Ahmad Abbasi Hajar Roshan-Shekalgourabi Dawood Hassanzadeh-Lelekaami 《Czechoslovak Mathematical Journal》2014,64(2):327-333
Let R be a commutative Noetherian ring with identity and I an ideal of R. It is shown that, if M is a non-zero minimax R-module such that dim Supp H I i (M) ? 1 for all i, then the R-module H I i (M) is I-cominimax for all i. In fact, H I i (M) is I-cofinite for all i ? 1. Also, we prove that for a weakly Laskerian R-module M, if R is local and t is a non-negative integer such that dim Supp H I i (M) ? 2 for all i < t, then Ext R j (R/I,H I i (M)) and Hom R (R/I,H I t (M)) are weakly Laskerian for all i < t and all j ? 0. As a consequence, the set of associated primes of H I i (M) is finite for all i ? 0, whenever dim R/I ? 2 and M is weakly Laskerian. 相似文献
4.
In this note we give a simple proof of the following result: Let R be a commutative Noetherian ring, an ideal of R and M a finite R-module, if H i (M) has finite support for all i < n, then Ass(H n (M)) is finite. 相似文献
5.
Let R be any ring. A right R-module M is called n-copure projective if Ext1(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext i (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension. 相似文献
6.
For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re
jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R
R(RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕
i=1
n
Mi is δ-coatomic if and only if each M
i (i=1,…, n) is δ-coatomic. 相似文献
7.
Muhammet Tamer Koşan 《Proceedings Mathematical Sciences》2009,119(4):453-458
Let R be a commutative Noetherian ring with non-zero identity and a be a maximal ideal of R. An R-module M is called minimax if there is a finitely generated submodule N of M such that M/N is Artinian. Over a Gorenstein local ring R of finite Krull dimension, we proved that the Socle of H
a
n
(R) is a minimax R-module for each n ≥ 0. 相似文献
8.
Amir Mafi 《Czechoslovak Mathematical Journal》2009,59(4):1095-1102
Let (R,m) be a complete local ring, a an ideal of R and N and L two Matlis reflexive R-modules with Supp(L) ⊆ V(a). We prove that if M is a finitely generated R-module, then Exti
R
i
(L, H
a
j
(M,N)) is Matlis reflexive for all i and j in the following cases:
In these cases we also prove that the Bass numbers of H
a
j
(M, N) are finite. 相似文献
(a) | dim R/a = 1 |
(b) | cd(a) = 1; where cd is the cohomological dimension of a in R |
(c) | dim R ⩽ 2. |
9.
Let (R, m) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R-module. In this paper it is shown that if p is a prime ideal of R such that dim R/p = 1 and (0:M p) is not finitely generated and for each i ? 2 the R-module Ext R i (M,R/p) is of finite length, then the R-module Ext R 1 (M, R/p) is not of finite length. Using this result, it is shown that for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext R i (N,M) are of finite length, if and only if, for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext R i (M,N) are of finite length. 相似文献
10.
Mohammad Reza Vedadi 《Mediterranean Journal of Mathematics》2012,9(1):143-151
A right R-module M is called co-Hopfian if injective endomorphisms of M
R
are surjective. It is shown that E(M
R
) is co-Hopfian if and only if M
R
does not contain an infinite direct sum
?i ? \mathbbNWi{{\oplus_{i \in \mathbb{N}}W_{i}}} of submodules such that each W
i+1 essentially embeds in W
i
. For many modules M
R
, including modules over a right FBN or right duo ring with Krull dimension, it is proved that E(M
R
) is co-Hopfian if and only if
(\mathbbN){(\mathbb{N})} ↪̸ M
R
for every non-zero X
R
. For a ring which has enough uniforms, the class of modules with co-Hopfian injective envelope is the same as the class of
modules with finite uniform dimension if and only if there are only finitely many isomorphism classes of indecomposable injective
modules. 相似文献
11.
We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z 2(R R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective. 相似文献
12.
Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D
R
(−) := Hom
R
(−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext
R
j
(R/a, D
R
(H
a
t
(M))) is finitely generated for all t > n and all j ⩾ 0, then we show that Hom
R
(R/a, D
R
(H
a
n
(M))) is also finitely generated. Specially, the set of prime ideals in Coass
R
(H
a
n
(M)) which contains a is finite.
Next, assume that (R, m) is a complete local ring. We study the finiteness properties of D
R
(H
a
r
(R)) where r is the least integer i such that H
a
r
(R) is not Artinian. 相似文献
13.
Kazem Khashyarmanesh 《Proceedings Mathematical Sciences》2010,120(1):35-43
Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom
R
(−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x
1, …, x
n
is a regular sequence on M contained in α, then H
(x1, …,xnR
n
D(H
a
n
(M))) is a homomorphic image of D(M), where H
b
i
(−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H
(x1, …,xn)R
n
(D(H
a
n
(M)))) ⋟ D(D(M)). 相似文献
14.
In this paper, we study Gorenstein injective modules over a local Noetherian ring R. For an R-module M, we show that M is Gorenstein injective if and only if Hom
R
(Ȓ,M) belongs to Auslander category B(Ȓ), M is cotorsion and Ext
i
R
(E,M) = 0 for all injective R-modules E and all i > 0.
Received: 24 August 2006 Revised: 30 October 2006 相似文献
15.
Robert M Guralnick 《Journal of Number Theory》1984,18(2):169-177
Let R be a Dedekind domain satisfying the Jordan-Zassenhaus theorem (e.g., the ring of integers in a number field) and Λ a module finite R-algebra. We extend classical results of Jacobinski, Roiter, and Drozd on orders and lattices. In particular, it is shown that the genus of a finitely generated Λ-module M is finite. Moreover, given M, there exist a positive integer t and a finite extension S of R such that a Λ-module N is the genus of M if and only if M(t) ? N(t) if and only if M ? S ? N ? S. 相似文献
16.
Hamidreza Rahmati 《Archiv der Mathematik》2009,92(1):26-34
A finite module M over a noetherian local ring R is said to be Gorenstein if Exti(k, M) = 0 for all i ≠ dim R. An endomorphism φ: R → R of rings is called contracting if for some i ≥ 1. Letting φR denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if HomR(φR, M) ≅ M and ExtiR(φR, M) = 0 for 1 ≤ i ≤ depth R.
Received: 7 December 2007 相似文献
17.
Claus Scheiderer 《Mathematische Zeitschrift》2010,266(1):1-19
Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules,
and the Mittag-Leffler ones. A right R-module M is called Baer if Ext1R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?R ?i ? IQi? ?i ? I(M?RQi){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Qi}i ? I{\{Q_i\}_{i\in I}} are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to
a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are
more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification
of the Mittag-Leffler modules. 相似文献
18.
Thomas Marley 《manuscripta mathematica》2001,104(4):519-525
Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H
i
I
(M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H
d
− 1
I
(M) has finite support for arbitrary R, I, and M.
Received: 31 October 2000 / Revised version: 8 January 2001 相似文献
19.
We consider two finitely generated graded modules over a homogeneous Noetherian ring $R = \oplus _{n \in \mathbb{N}_0 } R_n$ with a local base ring (R 0, m0) and irrelevant ideal R + of R. We study the generalized local cohomology modules H b i (M,N) with respect to the ideal b = b0 + R +, where b0 is an ideal of R 0. We prove that if dimR 0/b0 ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H b i (M,N)/a0 H b i (M,N) is Artinian, where $\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0$ ; for all i ≥ 0, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is asymptotically stable as n→?∞. Moreover, if H b i (M,N) n is a finitely generated R 0-module for all n ≤ n 0 and all j < i, where n 0 ∈ ? and i ∈ ?0, then for all n ≤ n 0, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is finite. 相似文献
20.
We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative
Noetherian local ring R and a reflexive R-module M such that ExtRi(M,R)=0 for all i>0, but ExtRi(M*,R)≠0 for all i>0.
Presented by Juergen Herzog
Mathematics Subject Classification (2000) 13D07. 相似文献