Artinian Modules Research Articles

Finitely Cogenerated Distributive Modules are Q-Cocyclic#‡

In this paper we consider a generalization of Vámos's proposition which asserts that finitely generated artinian and distributive modules are cyclic. Also, we show its duality as an answer to the problem which is shown by him (See Vámos, [Vámos, P. (1978). Finitely generated artinian and distributive modules are cyclic. Bull. London Math. Soc. 10(3):287–288]).

On the Countability of Noetherian Dimension of Modules#

It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ωω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.

Artinian level modules and cancellable sequences

In order to use dualization to study Hilbert functions of artinian level algebras we extend the notion of level sequences and cancellable sequences, introduced by Geramita and Lorenzini, to include Hilbert functions of certain artinian modules. As in the case of algebras a level sequence is cancellable, but now by dualization its reverse is also cancellable which gives a new condition on level sequences. We also give a characterization of the cancellable sequences involving Macaulay representations.

Modules with RD-Composition Series over a Commutative Ring

If R is a commutative ring,then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring,the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these numbers are equal if and only if M is a direct sum of cyclic modules. We also give an example of an artinian module over a noetherian domain,which has an RD-composition series with uniserial factors. Finally we prove that every pure-injective R-module is RD-injective if and only if R is an arithmetic ring.

Generalized Hopfian modules

A module is called generalized Hopfian (gH) if any of its surjective endomorphisms has a small kernel. Such modules are in a sense dual to weakly co-Hopfian modules that were defined and extensively studied in [A. Haghany, M.R. Vedadi, J. Algebra 243 (2001) 765–779]. Several equivalent formulations are given for gH modules and used in their study. Generalized Hopfian modules form a proper subclass of Dedekind finite modules, but this subclass is rather large in that it properly contains Hopfian, and in particular Noetherian modules, as well as Artinian modules and modules with finite dual Goldie dimension. For quasi-projective modules, the properties Dedekind finite, Hopfian and gH are all equivalent, yielding a ring R stably finite iff all finitely generated free R modules are gH. The gH property is shown to be a Morita invariant, and rings are characterized over which all finitely generated modules are gH. Some other classes of rings are also studied. These rings satisfy one of the conditions: all right ideals are gH; all annihilator right ideals are small (right domain); all left regular elements have small right annihilators (right directly finite). We will show: right domain ⇒ right directly finite ⇒ directly finite and these cannot be reversed. These results are by products of the study of special gH modules.

STRUCTURE OF THE FLAT COVERS OF ARTINIAN MODULES

The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form <TEX>$\prod_{p\epsilonAtt_R(A)}T-p$</TEX>, where <TEX>$Tp$</TEX> is the completion of a free <TEX>R$_{p}$</TEX>-module. Also, we construct a minimal flat resolution for R/xR-module 0: <TEX>$_AX$</TEX> from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = <TEX>$\chiA$</TEX>. This result leads to a description of the structure of a minimal flat resolution for <TEX>${H^n}_{\underline{m}}(R)$</TEX>, nth local cohomology module of R with respect to the ideal <TEX>$\underline{m}$</TEX>, over a local Cohen-Macaulay ring (R, <TEX>$\underline{m}$</TEX>) of dimension n.

On the Krull―Schmidt Theorem for Artinian Modules

A property of rings generalizing commutativity is introduced. If a ring satisfies this property, then the Krull--Schmidt theorem holds for Artinian modules over the ring. In particular, this property is fulfilled for local rings of finite rank and for rings such that their centers are surjectively mapped by the natural projection onto the factor with respect to the radical of the ring. A local ring for which the property fails is constructed; for the direct decompositions of Artinian modules over this ring there appear anomalies similar to the anomalies of direct decompositions of torsion-free Abelian groups of finite rank. Bibliography: 6 titles.

A Krull―Schmidt Theorem for Artinian Modules over Local Rings

The paper contains a generalization of Fitting's lemma for Artinian modules of arbitrary length over some non-Noetherian local rings. A counterexample to a strong form of the Krull―Schmidt theorem for Artinian modules over a local ring is constructed. Bibliography: 4 titles.

On the Loewy Length and the Noetherian Dimension of Artinian Modules

(2002). On the Loewy Length and the Noetherian Dimension of Artinian Modules. Communications in Algebra: Vol. 30, No. 3, pp. 1077-1084.

Local Duality for Modules over Noetherian Commutative Rings

Some applications of the general theorem on the existence of local duality for modules over Noetherian commutative rings are given. Let $$\Lambda$$ be a Noetherian commutative ring, let $$\mathcal{M} = \left\{ \mathfrak{M} \right\}$$ be a set of maximal ideals in $$\Lambda$$ , and let $$\widehat\Lambda = \underleftarrow {\lim }\,\Lambda _\mathfrak{M} ,\Gamma \left( \Lambda \right) = \prod\limits_{\mathfrak{M} \in \mathcal{M}} {\widehat\Lambda _\mathfrak{M} }$$ . Then the category of Artinian modules is dual to the category of Noetherian modules. Several structural results are proved, including the theorem on the structure of Artinian modules over principal ideal domains. For rings of special kinds, double centralizer theorems are proved. Bibliography: 5 titles.

The I-adic completion and local homology for Artinian modules

Let I be an ideal of a commutative ring R and M an R-module. It is well known that the I-adic completion functor ΛI defined by ΛI(M) = lim←tM/ItM is an additive exact covariant functor on the category of finitely generated R-modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, ΛI is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence of left derived functors {LIi} of ΛI, in which LI0 is right exact, but in general LI0 ≠ ΛI. Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R-regular sequence, Matlis proved in [9, 10] thatwhere D(−) = HomR(−; E(R/[mfr ])) is the Matlis dual functor, and thatIn [18, 5] A.-M. Simon shows that LIo(M) = M and LIi(M) = 0 for i &gt; 0, provided that M is complete with respect to the I-adic topology.Later, Greenlees and May [3] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R) bywhere x is a finitely generated system of I. Then they showed, under some conditions on x which are satisfied when R is Noetherian, that HI[bull ](M) ≅ LI[bull ](M). Recently, Tarrío, López and Lipman [1] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.

CHAIN CONDITIONS ON QUOTIENT FINITE DIMENSIONAL MODULES

This paper is motivated by a recent work of C. Faith [9] who has proved that a quotient finite dimensional module which satisfies the ascending chain condition on subdirectly irreducible submodules is Noetherian. A natural question to ask (also raised by Faith [11]) is whether its dual holds true. We answer this in the affirmative, and provide a characterization of Artinian modules dual to Faith's Theorem. We also extend these results to the more general settings of dual Krull dimension and Krull dimension respectively.

Krull–Remak–Schmidt Fails for Artinian Modules over Local Rings

Let R be a ring. Any R-module M which is Artinian or Noetherian can be written as the direct sum of a finite number of indecomposable R-modules. The theorem of Krull–Remak–Schmidt asserts that in the case where M is of finite length, such a decomposition is unique up to isomorphism. On the other hand, examples of Noetherian R-modules which have essentially different decompositions have been known for a long time. The first examples of Artinian R-modules with essentially different decompositions were published only in 1995 by Facchini, Herbera, Levy and Vamos. In order to construct such examples, one needs to deal with suitable rings R. Note that for R Noetherian or commutative, all the Artinian modules have the Krull–Remak–Schmidt property. In 1998, Facchini raised the problem of whether the same is true in the case where R is a local ring. The aim of this note is to show that this is not so: we are going to present a local ring R and Artinian R-modules M with essentially different direct decompositions into indecomposables. The military importance of these results has been discussed during the NATO meeting at Constantia (August 2000) which was organized by K. W. Roggenkamp.

A note on the dual of Burch's inequality

The aim of this paper is to improve the main result of [5] (Theorem 3.5). We show that if A is a non-zero Artinian module over a commutative ring R and a⊆ b are ideals of R such that (0 : A b) ≠ 0 , then the dual of Burch's inequality S b ( a,A) ≤ Kdim R(A)− width b (0 : A a i) (i ⪢ 0) holds (the dual notions S b ( a,A), Kdim R(A), width b (A) are explained in [5]).

The Theory of Local Homology for Artinian Modules

The Theory of Local Homology for Artinian Modules

The Theory of Local Homology for Artinian Modules

In this paper, the theory of local homology for Artinian modules is developed, which in some aspect is similar to the theory of local cohomology. There are given some criteria for local homology modules to be “good” and dual of the ideal transform functor is studied.

Koassoziierte Primideale in 2-Dimensionalen Regulären Ringen

Suppose that R is a 2-dimensional regular local ring, pa prime ideal of height 1, and u = r/s an element of the quotient field of R. We give criteria for pbeing coassociated to the R-module B = R[u], i.e., that there is an Artinian factor module B/X such that p= AnnR(B/X). These criteria are formulated using the ideal a= (r, s) and the intersection numbers σ(αi, r), σ(αi, s), where α1, ... αh are the branches of pin the completion R.

Strong and weak F -regularity are equivalent for graded rings

It is shown that the tight closure of a submodule in a Artinian module is the same as its finitistic tight closure, when the modules are graded over a finitely generated [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-graded ring over a perfect field. As a corollary, it is deduced that for such a graded ring, strong and weak F -regularity are equivalent. As another application, the following conjecture of Hochster and Huneke is proved: Let ( R, m ) be a finitely generated [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-graded ring over a field with unique homogeneous maximal ideal m , then R is (weakly) F -regular if and only if R m is (weakly) F -regular.

Co-cohen-macaulayness of certain graded artinian modules

Co-cohen-macaulayness of certain graded artinian modules

Direct decompositions in Artinian modules over FC-hypercentral groups

Direct decompositions in Artinian modules over FC-hypercentral groups