Rings Of Finite Representation Type Research Articles

ℵ0-Distributive modules and rings

Let A be a ring with minimum condition on principal right ideals. It is proved that countably distributive right (left) A-modules coincide with Artinian (Noetherian) right (left) A-modules. Rings, over which all right modules are [Formula: see text]-distributive coincide with rings of finite representation type. Rings, whose right modules are semidistributive, coincide with Kawada rings, over basis rings of which all right modules are completely cyclic.

Pure semisimplicity conjecture and Artin problem for dimension sequences

Inspired by a recent paper due to José Luis García, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight , of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding F ↪ M n ( G ) exists provided that n < 5 . As a byproduct, we obtain a division ring extension G ⊆ F such that the bimodule F F G has the right dimension sequence ( 1 , 2 , 2 , 2 , 1 , 4 ) . Finally, we formulate Conjecture A , which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture.

Rings whose pure-injective right modules are direct sums of lifting modules

It is shown that every pure-injective right module over a ring R is a direct sum of lifting modules if and only if R is a ring of finite representation type and right local type. In particular, we deduce that every left and every right pure-injective R-module is a direct sum of lifting modules if and only if R is (both sided) serial artinian. Several examples are given to show that this condition is not left–right symmetric.

On the endofiniteness of a key module over pure semisimple rings

Let R R be a left pure semisimple ring such that there are no non-zero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R R -mod, and let W W be the left key R R -module; i.e., W W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R R -modules. We show that if the module W W is endofinite, then R R is a ring of finite representation type. This settles a question considered in [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007), 361-376] for hereditary rings.

The structure of locally finite varieties with polynomially many models

We prove that a locally finite variety has at most polynomially many (in k k ) non-isomorphic k k –generated algebras if and only if it decomposes into a varietal product of an affine variety over a ring of finite representation type, and a sequence of strongly Abelian varieties equivalent to matrix powers of varieties of H H -sets, with constants, for various finite groups H H .

A new characterisation of hereditary pi rings

In this paper we prove a new characterisation of hereditary PI rings, namely we show that a Noetherian, but not Artinian, PI ringR that is an order in an Artinian ring splits into a direct sum of an Artinian ring of finite representation type and hereditary semiprime rings if and only if all its proper Artinian factor rings are of finite representation type. We also show, through examples, that the above characterisation does not hold for some more general settings.

ON SELF-DUALITY OF AUSLANDER RINGS OF LOCAL SERIAL RINGS

Let A be an artinian ring of finite representation type. A ring R is called the right Auslander ring of A if it is an endomorphism ring of the direct sum of modules in a complete set of represent a...

Relations for relative Grothendieck groups of rings of finite representation type

For a ring R of finite representation type the set of free generators is described for the group of relations of the relative Grothendieck group K0(R,F) corresponding to an additive subfunctor F of the functorExt 1 R, which yields an extension to the absolute case of the corresponding results of Butler and Auslander. Bibliography:8 titles.

Modules with Indecomposable Decompositions That Complement Maximal Direct Summands

Generalizing a fundamental property of semisimple modules, Anderson w x and Fuller 1 introduced the following important concept for direct decompositions of modules. A decomposition M s [ M is said to ig I i complement direct summands in case for each direct summand A of M, Ž . Ž w there is a subset H : I such that M s A [ [ M cf. also 2, p. ig H i x. 141 . Such a decomposition is necessarily an indecomposable decomposition, and it is an interesting problem to study under which conditions an indecomposable decomposition complements direct summands. There is w x an extensive literature concerning this problem. By 1, Theorem 6 , right perfect rings can be characterized by the property that all projective right modules have decompositions that complement direct summands. Fuller w x 7 showed that over a generalized uniserial ring every module has a decomposition that complements direct summands, thus providing a first class of non-semisimple rings satisfying this property. More generally, w x Ž . Tachikawa 14 proved that every left and right module over a ring of finite representation type has a decomposition that complements direct summands, and that the converse is also true was established by Fuller and w x w x Reiten 9 . Restricting just to one side, Fuller 8 showed that rings over which every right module has a decomposition that complements direct summands are precisely the rings over which every right module is a direct Ž sum of finitely generated modules they are also called right pure-semisim-

Subrings of I-rings and S-rings

LetRbe a non-commutative associative ring with unity1≠0, a leftR-module is said to satisfy property (I) (resp. (S)) if every injective (resp. surjective) endomorphism ofMis an automorphism ofM. It is well known that every Artinian (resp. Noetherian) module satisfies property (I) (resp. (S)) and that the converse is not true. A ringRis called a left I-ring (resp. S-ring) if every leftR-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is known that a subringBof a left I-ring (resp. S-ring)Ris not in general a left I-ring (resp. S-ring) even ifRis a finitely generatedB-module, for example the ringM3(K)of3×3matrices over a fieldKis a left I-ring (resp. S-ring), whereas its subringB={[α00βα0γ0α]/α,β,γ∈K}which is a commutative ring with a non-principal Jacobson radicalJ=K.[000100000]+K.[000000100]is not an I-ring (resp. S-ring) (see [4], theorem 8). We recall that commutative I-rings (resp S-tings) are characterized as those whose modules are a direct sum of cyclic modules, these tings are exactly commutative, Artinian, principal ideal rings (see [1]). Some classes of non-commutative I-rings and S-tings have been studied in [2] and [3]. A ringRis of finite representation type if it is left and right Artinian and has (up to isomorphism) only a finite number of finitely generated indecomposable left modules. In the case of commutative rings or finite-dimensional algebras over an algebraically closed field, the classes of left I-rings, left S-rings and rings of finite representation type are identical (see [1] and [4]). A ringRis said to be a ring with polynomial identity (P. I-ring) if there exists a polynomialf(X1,X2,…,Xn),n≥2, in the non-commuting indeterminatesX1,X2,…,Xn, over the centerZofRsuch that one of the monomials offof highest total degree has coefficient1, andf(a1,a2,…,an)=0for alla1,a2,…,aninR. Throughout this paper all rings considered are associative rings with unity, and by a moduleMover a ringRwe always understand a unitary leftR-module. We useMRto emphasize thatMis a unitary rightR-module.

Special Quasi-triads and Integral Group Rings of Finite Representation Type, I

In this paper, we investigate a particular order over a complete discrete valuation ring which we define to be a special quasi-triad. Such orders manifest themselves in several areas, most notably, the study of orders with finitely many non-isomorphic, indecomposable lattices. We are able to classify all the indecomposable lattices over such an order and thereby determine which such special quasi-triads have the property that every indecomposable lattice is isomorphic to a right ideal. This paper is necessary for the classification of lattices over integral group rings of finite type, which is done in another paper by the authors

Special Quasi-triads and Integral Group Rings of Finite Representation Type, II

We describe the structure of the integral group ring ZG, when G is a group with cyclic Sylow subgroups, as a subdirect sum of hereditary orders in cyclic crossed product algebras. In the case where ZG is of finite representation type, we deduce the structure of all genera of ZG-lattices. Our principal applications are the following. (i) We determine which groups G have the property that each ZG-lattice is isomorphic to a direct sum of right ideals. (ii) We determine which groups G with cyclic Sylow subgroups have the property that each ZG-lattice has a unique number of indecomposable summands. (iii) We show that, for groups G with cyclic Sylow subgroups, the ring structure of the rational group algebra QG determines the group G up to isomorphism.

Matrix Reductions for Artinian Rings, and an Application to Rings of Finite Representation Type

Matrix Reductions for Artinian Rings, and an Application to Rings of Finite Representation Type

Commutative group rings of finite representation type

Let Z p denote the localization (but not the completion) of the integers at the prime p. Then, for finite groups G, the group ring Z p G has finite representation type if and only if the p-Sylow subgroups of G are cyclic of order at most p 2. In this paper, we determine the possible ranks of indecomposable Z p G-lattices for finite abelian groups G for which Z p G has finite representation type. In particular, for such groups G, we show that every indecomposable Z p G-lattice can be embedded as a sublattice of Z p G (4), but not, in general, as a sublattice of Z p G (3).

Burnside rings of finite representation type

Let G be a finite group. It is proved that the localised Buinside ring Ωp(G) is of finite representation type if and only if for each p–perfect subgroup H of G, , where means the conjugacy class of K.

The classification of homogeneous Cohen-Macaulay rings of finite representation type

In this paper we classify the homogeneous Cohen-Macaulay rings which are of finite representation type, that is, the Cohen-Macaulay rings which are positively graded and generated in degree 1, with an algebraically closed field of characteristic 0 in degree 0, and which have, up to isomorphisms and shifts in the grading, only a finite number of indecomposable maximal Cohen-Macaulay modules (MCM-modules). Our main contribution is to show that a homogeneous Cohen-Macaulay ring of finite representation type and dimension >__ 2 must have minimal multiplicity. Putting this together with previous results of Auslander, Auslander-Reiten, Buchweitz-Greuel-Schreyer, Greuel-Kn6rrer, and Solberg, we obtain:

A note on indecomposable modules

In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinian.

Hereditary artinian rings of finite representation type and extensions of simple artinian rings

In [1], Dowbor, Ringel and Simson consider hereditary artinian rings of finite representation type; it is known that if A is an hereditary artinian algebra of finite representation type, finite-dimensional over a field, then it corresponds to a Dynkin diagram in a natural way; they show that an hereditary artinian ring of finite representation type corresponds to a Coxeter diagram. However, in order to construct an hereditary artinian ring of finite representation type corresponding to a Coxeter diagram that is not Dynkin, they show that it is necessary though not sufficient to find an extension of skew fields such that the left and right dimensions are both finite but are different. No examples of such skew fields were known at the time. In [3], I constructed such extensions, and the main aim of this paper is to extend the methods of that paper to construct an extension of skew fields having all the properties needed to construct an hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5).

Artin's problem for skew field extensions

For a commutative field extension, L ⊃ K, it is clear that a left basis of L over K; is also a right basis of L over K; however, for an extension of skew fields, this may easily fail, though it is hard to determine whether the right and left dimension may be different. Cohn ([4], ch. 5), however, was able to find extensions of skew fields such that the left and right dimensions were an arbitrary pair of cardinals subject only to the restrictions that neither were 1 and at least one of them was infinite. In this paper, I shall present a new approach that allows us to construct extensions of skew fields such that the left and right dimensions are arbitrary integers not equal to 1. In a subsequent paper, [7], I shall present related results and consequences; in particular, there is a construction of a hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5) answering the question raised by Dowbor, Ringel and Simson[5].

Rings of finite representation type and modules of finite Morley rank

This paper deals with rings whose modules have very pleasant decomposition properties: the right pure semisimple rings. The category of right modules over such a ring is, in many ways, similar to the category of modules over a semisimple artinian ring. Yet it is still an open question whether or not pure semisimplicity actually is a two-sided property (and hence [30] coincides with being of finite representation type). Most of the arguments in this paper are model-theoretic-algebraic in nature: it seems that ideas from model theory, and from the model theory of modules, find natural application in the context considered here. The paper is organized into three sections. The first is introductory and with it I attempt to make the other sections accessible to a reasonably wide readership. In the second section are gathered together various equivalents to right pure semisimplicity. Their equivalence is given a comparatively short, unified, essentially model-theoretic-algebraic proof. The main theorem, in the third section, is that a ring is of finite representation type if and only if all its (right) modules have finite Morley rank. A number of related results are developed. Most of these involve some model theory in their statement, but many are, in essense, algebraic. Since a good deal of information has been compressed into the introductory section, this section should, perhaps, first be read through quickly, and then referred back to as the need arises. This section also is meant to serve as background to a sequel, in which I consider pp formulas and types in terms of the corresponding matrices, and in which the main aim is more purely algebraic. The global conventions are: R denotes a ring with identity; “module”