Modules Of Finite Length Research Articles

On rings whose modules of finite length are all cyclic

We give some characterisations of rings R whose modules with composition series are all cyclic. In particular, we prove that all left R-modules of finite length are cyclic if and only if R has no nonzero Artinian factor rings.

Modules of finite length and finite projective dimension

Modules of finite length and finite projective dimension

On local right pure semisimple rings of length two or three

We investigate in this paper non-commutative local rings of the smallest length that are potential counter-examples to the pure semisimplicity conjecture. Throughout the paper is an associative ring with an identity element. We call local, if the Jacobson radical ( ) of is a two-sided maximal ideal. We denote by mod( ) the category of finitely generated right -modules. Given a right -module of finite length we denote by ( ) the length of . We recall that a ring is said to be of finite representation type, if is artinian and the number of the isomorphism classes of finitely generated indecomposable right (and left) -modules is finite. Following [24] we call a ring right pure semisimple, if every right -module is a direct sum of finitely generated modules. It is well known that a ring is of finite representation type if and only if is right pure semisimple and is left pure semisimple (see [2], [11], [18], [22]–[24]). It is still an open question, called the pure semisimplicity conjecture, if a right pure semisimple ring is of finite representation type (see [2] and [24], [25], [28]). In [13] the question is answered in affirmative for rings satisfying a polynomial identity and for self-injective rings (see also [7], [19] and [31]). The reader is referred to [42] and to the author’s expository papers [30] and [32] for a basic background and historical comments on the pure semisimplicity conjecture. It was shown by the author in [28] and [33] that there is a chance to find a counter-example to the pure semisimplicity conjecture and might be hereditary with two simple non-isomorphic modules. The existence of a counter-example depends on a generalized Artin problem on division ring extensions. In the present paper we are mainly interested in the existence of counter-examples to pure semisimplicity conjecture that are local of the smallest length, that is, of length ( ) two or three. This continues our study started in [28], [35] and [33]. It is shown in Lemma 3.1 that every such a local ring has ( )2 = 0. Therefore we study representation-infinite right pure semisimple local rings with ( )2 = 0 such that the Auslander-Reiten quiver (mod ) is of the form · · · → •→ •→ •→

A family of noetherian rings with their finite length modules under control

We investigate the category modΛ of finite length modules over the ring Λ=A ⊗ k Σ, where Σ is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E j gives rise to a quasiprogenerator P j = A ⊗ E j . By a result of K. Fuller, P j induces a category equivalence from which we deduce that modΛ ≃ ∐j mod EndP j . As a consequence we can (1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring Λ such that modA ≃ modΛ (2) find twisted versions Λ of algebras of wild representation type such that Λ itself is of finite or tame representation type (in mod) (3) describe for certain rings Λ the minimal almost split morphisms in modΛ and observe that almost all of these maps are not almost split in ModΛ.

THE LENGTH OF NOETHERIAN MODULES

ABSTRACT We define an ordinal valued length for Noetherian modules which extends the usual definition of composition series length for finite length modules. Though originally defined by Gulliksen[7] in the 1970s, this extension has seldom been used in subsequent research. Despite this neglect, we will show that the ordinal valued length is a quite natural measure of the size of a Noetherian module, and has advantages over more familiar measures such as uniform dimension, Krull dimension, and reduced rank. We will also demonstrate how some familiar properties of left Noetherian rings can be proved efficiently using length and the arithmetic properties of ordinal numbers.

Pure Injective Envelopes of Finite Length Modules over a Generalized Weyl Algebra

We investigate certain pure injective modules over generalised Weyl algebras. We consider pure injective hulls of finite length modules, the elementary duals of these, torsionfree pure injective modules, and the closure in the Ziegler spectrum of the category of finite length modules supported on a nondegenerate orbit of a generalized Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrasts with the behaviour of pure injective modules over finite-dimensional algebras and hereditary orders.

THE FINITE LENGTH MODULES FOR THIN -GRADED RINGS

A -graded ring R = is called thin, provided R 0 = K is a skew field and each K–K-bimodule Ri has left and right dimension at most 1. Such rings occur, for example, as fiber products of skew polynomial rings, as numerical skew monoid rings or as factor rings or localizations of generalized Weyl algebras. We show that in case R is (graded) affine with no homogeneous units in nonzero degree, the representation type (finite, “twisted tame infinite,” “twisted wild”) can be read off from the combinatorial structure, that is, from the vanishing of products of homogeneous elements of certain degrees. Those rings R which are of twisted wild representation type admit by our definition a representation embedding from the category of finite dimensional modules over some twisted wild algebra T(K, σ K ⊕τ K) into the category mod R of finite length R-modules. The rings which we call here “twisted tame” come close to twisted versions of string algebras, for which we obtain the classification of the indecomposable finite length modules as a version of the Butler–Ringel theorem. Finally, for rings which are either not affine or which have homogeneous units in nonzero degree, the representation type does not depend on the combinatorial structure alone, as examples show.

Ideals attaining a given Hilbert function

We improve the result of Charalambous and Evans [C-E] to show that the Betti number sequence in their example of incomparable minimals among the resolutions for a fixed Hilbert function is indeed minimal. Their example was dependent upon the graded betti numbers. We give an example of a finite length Hilbert function and two cyclic finite length modules attaining the Hilbert function for which the betti number sequences are incomparable, i.e., independent of the grading.

Finite Length and Pure-Injective Modules over a Ring of Differential Operators

Let k be an algebraically closed field of characteristic zero, On=k[[x1,…,xn]] the ring of formal power series over k, and Dn the ring of differential operators over On. Suppose that ρ is a prime ideal of On of height n−1; i.e., A=On/ρ is a curve. We prove that every indecomposable finite length module over Dn with support on ρ is uniserial with isomorphic or alternating composition factors. For the ring D(A) of differential operators over A we also classify indecomposable pure-injective modules and show that the Cantor–Bendixson rank of the Ziegler spectrum over D(A) is equal to 2.

Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules F such that direct sums of F-covers (F-envelopes) are F-covers (F-envelopes), F-covering (Fenveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of F-covers are F-covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.

On injective hulls of simple modules

We characterize a ring over which every left module of finite length has an injective hull of finite length. Using this, we show that finite normalizing extensions of such a ring also have the same property. We also consider rings having the property that the injective hull of every simple module is artinian. We show that certain noncommutative noetherian rings have this property.

Betti numbers of modules of essentially monomial type

Let R be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of R-modules of essentially monomial type. These modules are defined with respect to various R-regular sequences. For example, finite length modules of monomial type over regular local rings of dimension n are modules of essentially monomial type with respect to R-regular sequences of length n. If a module is of essentially monomial type with respect to an R-regular sequence of length n, then the rank of its i-th syzygy is at least (MDl) and its i-th Betti number is at least (n).

A Frobenius characterization of finite projective dimension over complete intersections

Let M be a module of finite length over a complete intersection (R,m) of characteristic \(p>0\). We characterize the property that M has finite projective dimension in terms of the asymptotic behavior of a certain length function defined using the Frobenius functor. This may be viewed as the converse to a theorem of S. Dutta. As a corollary we get that, in a complete intersection (R,m), an m-primary ideal I has finite projective dimension if and only if its Hilbert-Kunz multiplicity equals the length of R/I.

Left almost split morphisms and generalized weyl algebras

It is proved that the category of modules of finite length over a broad class of generalized Weyl algebras contains no left almost split morphism starting from a simple module. It is shown that a similar assertion holds for the algebraUsl2(k) over an algebraically closed field k of characteristic 0. As a by-product, a new series of simple modules for such algebras is constructed.

Some model theory over hereditary noetherian domains

Questions in the model theory of modules over hereditary noetherian domains are investigated with particular attention being paid to differential polynomial rings and to generalized Weyl algebras. We prove that there exists no isolated point in the Ziegler spectrum over a simple hereditary generalized Weyl algebra A of the sort considered by Bavula [ Algebra i Analiz 4 (1) (1992), 75–97] over a field k with char( k ) = 0 (the first Weyl algebra A 1 ( k ) is one such) and the category of finite length modules over A does not have any almost split sequence. We show that the theory of all modules over a wide class of generalized Weyl algebras and related rings interprets the word problem for groups, and in the case that the field is countable there exists a superdecomposable pure-injective module over A . This class includes, for example, the universal enveloping algebra Usl 2 ( k ).

A note on self-dual modules and Dedekind rings

It is shown that a commutative noetherian ring is a finite direct sum of Dedekind rings and artinian uniserial rings if and only if every module of finite length is selfdual. A module of finite length is said to be selfdual if it is isomorphic to its dual with respect to the minimal injective cogenerator.

An Axiomatic Description of a Duality for Modules

Take for instance a k-algebra 4 over a field k. Then the usual k-dual DM=Homk(M, k) of a finite dimensional 4-module M satisfies all three properties. However for infinite dimensional 4-modules the condition (D1) does not hold, and for most rings there is even on the level of the finite length modules no duality functor between right and left modules available. The aim of this note is to describe without any restriction on the ring 4 a bijection between certain classes of right and left 4-modules which satisfies (D1) (D3) and which is uniquely determined by these properties. This bijection covers the k-duality mentioned above but usually there will be non-finitely generated modules on which D is defined. Our approach gives a new interpretation of a construction which is known as elementary duality amongst model theorists, and which is due to Herzog [6], see also [9]. We formulate now the main result. To this end denote by Mod(4) the category of (right) 4-modules and by mod(4) the full subcategory of all finitely presented 4-modules. We shall identify the category of left 4-modules with Mod(4). Recall that a 4-module M is pure-injective if every pure monomorphism starting in M splits. A map .: M N is a pure monomorphism if . 4 X : M 4 X N 4 X is a monomorphism for every X in mod(4). We call an indecomposable pure-injective 4-module article no. AI971660

Betti Numbers of Modules of Exponent Two over Regular Local Rings

Let (R,m,K) be a regular local ring of dimensionnand letMbe a finite length module overR. This paper gives an affirmative answer to Horrocks' questions whenm2M=0, that is, in this case the rank of theith syzygy ofMis at least[formula]and theith Betti number ofMis at least[formula].

On modules of finite length

It is shown that for a finitely generated self-projective right R-module M, if every finitely generated module in α[M]is a direct sum of an M-projective module and a quasi-continuous module, then M is a module of finite length. In particular, R is a right artinian ring if every finitely generated right R-module is a direct surn of a projective module and a quasi-continuous module.

A dichotomy for finite length modules induced by local duality

A dichotomy for finite length modules induced by local duality