Gorenstein injective modules and Auslander categories

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     

On the Matlis duals of local cohomology modules

Let ( ) be a commutative Noetherian local ring with non-zero identity, an ideal of R and M a finitely generated R-module with . Let D(–) := Hom _R (–, E) be the Matlis dual functor, where is the injective hull of the residue field . We show that, for a positive integer n, if there exists a regular sequence and the i-th local cohomology module H ⁱ _a (M) of M with respect to is zero for all i with i > n then The author was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 85130023). Received: 9 August 2006     

Finiteness properties of minimax and coatomic local cohomology modules

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.     

Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).     

Spectral Exponent of Finite Sums of Weighted Positive Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Spaces

In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form

Independence of the Total Reflexivity Conditions for Modules

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 Ext_Rⁱ(M,R)=0 for all i>0, but Ext_Rⁱ(M^*,R)≠0 for all i>0. Presented by Juergen Herzog Mathematics Subject Classification (2000) 13D07.     

Generalizations of coatomic modules

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 M_i is δ-coatomic if and only if each M _i (i=1,…, n) is δ-coatomic.     

Über die Bedingung Going up für {R \subset \hat{R}}

Let (R,\mathfrak m){(R,\mathfrak m)} be a noetherian, local ring with completion [^(R)]{\hat{R}} . We show that R ì [^(R)]{R \subset \hat{R}} satisfies the condition Going up if and only if there exists to every artinian R-module M with Ann_R(M) ì \mathfrakp{{\rm Ann}_R(M) \subset \mathfrak{p}} a submodule U ì M{U \subset M} with Ann_R(U)=\mathfrakp.{{\rm {Ann}}_R(U)=\mathfrak{p}.} This is further equivalent to R being formal catenary, to α(R) = 0 and to H^d_{\mathfrakq/\mathfrakp}(R/\mathfrakp)=0{H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0} for all prime ideals \mathfrakp ì \mathfrakq \subsetneq \mathfrakm{\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}} where d = dim(R/\mathfrakp){d = {\rm {dim}}(R/\mathfrak{p})}.     

Artinian cofinite modules over complete Noetherian local rings

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.     

On the prime radical and Baer’s lower nilradical of modules

Let M be a left R-module. In this paper a generalization of the notion of m-system set of rings to modules is given. Then for a submodule N of M, we define := { m ε M: every m-system containing m meets N}. It is shown that is the intersection of all prime submodules of M containing N. We define rad _R (M) = . This is called Baer-McCoy radical or prime radical of M. It is shown that if M is an Artinian module over a PI-ring (or an FBN-ring) R, then M/rad _R (M) is a Noetherian R-module. Also, if M is a Noetherian module over a PI-ring (or an FBN-ring) R such that every prime submodule of M is virtually maximal, then M/rad _R (M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either rad _R (M) = M or rad _R (M) = ∩ _i=1 ⁿ for some maximal ideals of R. Also, Baer’s lower nilradical of M [denoted by Nil_* ( _R M)] is defined to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, rad _R (M) = Nil_*( _R M) and, for any module M over a left Artinian ring R, rad _R (M) = Nil_*( _R M) = Rad(M) = Jac(R)M. This research was in part supported by a grant from IPM (No. 85130016). Also this work was partially supported by IUT (CEAMA). The author would like to thank the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.     

Weighted sums of squares in local rings and their completions, I

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 Ext¹_R (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 ? I}Q_i? ?_{i ? I}(M?_RQ_i){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Q_i}_{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.     

A generalization of the finiteness problem of the local cohomology modules

Let R be a commutative Noetherian ring and \(\mathfrak{a}\) an ideal of R. We introduce the concept of \(\mathfrak{a}\) -weakly Laskerian R-modules, and we show that if M is an \(\mathfrak{a}\) -weakly Laskerian R-module and s is a non-negative integer such that Ext _R ^j \((R/\mathfrak{a},H_\mathfrak{a}^i (M))\) is \(\mathfrak{a}\) -weakly Laskerian for all i < s and all j, then for any \(\mathfrak{a}\) -weakly Laskerian submodule X of \(H_\mathfrak{a}^s (M)\) , the R-module \(Hom_R (R/\mathfrak{a},H_\mathfrak{a}^s (M)/X)\) is \(\mathfrak{a}\) -weakly Laskerian. In particular, the set of associated primes of \(H_\mathfrak{a}^s (M)/X\) is finite. As a consequence, it follows that if M is a finitely generated R-module and N is an \(\mathfrak{a}\) -weakly Laskerian R-module such that \(H_\mathfrak{a}^i (N)\) (N) is \(\mathfrak{a}\) -weakly Laskerian for all i < s, then the set of associated primes of \(H_\mathfrak{a}^s (M,N)\) (M,N) is finite. This generalizes the main result of S. Sohrabi Laleh, M.Y. Sadeghi, and M.Hanifi Mostaghim (2012).     

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

On connectedness and indecomposibility of local cohomology modules

Let I denote an ideal of a local Gorenstein ring . Then we show that the local cohomology module , c = height I, is indecomposable if and only if V(I _d ) is connected in codimension one. Here I _d denotes the intersection of the highest dimensional primary components of I. This is a partial extension of a result shown by Hochster and Huneke in the case I the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of is a local Noetherian ring if dim R/I  =  1.     

Model theory of the regularity and reflection schemes

Abstract  This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here is a language with a distinguished linear order <, and REF consists of formulas of the form

Bass Numbers of Local Cohomology Modules with Respect to an Ideal

Let be a commutative Noetherian local ring and let be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to . As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module are finite if one of the following holds:

A geometrical problem arising in a signal restoration algorithm

Let ℤ_2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ_2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ_2N→ℝ is given by φ_f(k)=arg for k ∈ ℤ_2N, where denotes the discrete Fourier transforms of f. For a fixed s∈L let K_s denote the cone of all f:ℤ_2N→ℝ which satisfy φ_f ≡ φ_s and let M_s be its linear span. The angle α_s between M_s and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove the following conjectures of Urieli et al. [7] who verified them for the N≤3 case:

GRADE AND GORENSTEIN DIMENSION

Let R be a commutative Noetherian ring and let M be a finite (that is, finitely generated) R-module. The notion grade of M, grade M, has been introduced by Rees as the least integer t ≥ 0 such that Ext ^t _R (M,R) ≠ 0, see [11] Rees, D. 1957. The Grade of an Ideal or Module. Proc. Camb. Phil. Soc., 53: 28–42. [Crossref] , [Google Scholar]. The Gorenstein dimension of M, G-dim M, has been introduced by Auslander as the largest integer t ≥ 0 such that Ext ^t _R (M, R) ≠ 0, see [3] Auslander, M. 1967. Anneaux De Gorenstein Et Torsion En Algebre Commutative Edited by: Mangeney, M., Peskine, C. and Szpiro, L. Paris: Ecole Normale Superieure de Jeunes Filles.  [Google Scholar]. In this paper the R-module M is called G-perfect if grade M = G-dim M. It is a generalization of perfect module. We prove several results for the new concept similar to the classical results.     

Algebraic representation of mappings between submodule lattices

We show that under certain weak conditions on the module _R M, every mapping