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

1987 ◽  
Vol 102 (3) ◽  
pp. 411-420 ◽  
Author(s):  
Aidan Schofield
Keyword(s):  
Artinian Ring ◽  
Representation Type ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Artinian Rings ◽  
Finite Dimensional ◽  
Coxeter Diagram ◽  
Skew Fields ◽  
Left And Right ◽  
Natural Way

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).

scholarly journals Algebras of bounded finite dimensional representation type

1995 ◽  
Vol 37 (3) ◽  
pp. 289-302 ◽  
Author(s):  
Allen D. Bell ◽  
K. R. Goodearl
Keyword(s):  
Representation Type ◽  
Dimensional Representation ◽  
Dimensional Algebra ◽  
Finite Dimensional Representation ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Left And Right ◽  
Bounded Representation ◽  
Algebra Over A Field

It is well known that for finite dimensional algebras, “bounded representation type” implies “finite representation type”; this is the assertion of the First Brauer-Thrall Conjecture (hereafter referred to as Brauer-Thrall I), proved by Roiter [26] (see also [23]). More precisely, it states that if R is a finite dimensional algebra over a field k, such that there is a finite upper bound on the k-dimensions of the finite dimensional indecomposable right R-modules, then up to isomorphism R has only finitely many (finite dimensional) indecomposable right modules. The hypothesis and conclusion are of course left-right symmetric in this situation, because of the duality between finite dimensional left and right R-modules, given by Homk(−, k). Furthermore, it follows from finite representation type that all indecomposable R modules are finite dimensional [25].

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

Artin's problem for skew field extensions

1985 ◽  
Vol 97 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. H. Schofield
Keyword(s):  
Artinian Ring ◽  
Field Extension ◽  
Subsequent Paper ◽  
Commutative Field ◽  
Finite Representation ◽  
New Approach ◽  
Skew Field ◽  
Skew Fields ◽  
Left And Right ◽  
The Right

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].

PURE SEMISIMPLE FINITELY ACCESSIBLE CATEGORIES AND HERZOG'S CRITERION

2007 ◽  
Vol 06 (06) ◽  
pp. 1001-1025 ◽  
Author(s):  
A. I. CÁRCELES ◽  
J. L. GARCÍA
Keyword(s):  
Sufficient Conditions ◽  
Representation Type ◽  
Locally Finite ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Artinian Rings ◽  
Semisimple Ring ◽  
Accessible Categories ◽  
Accessible Category ◽  
Necessary And Sufficient

Let [Formula: see text] be a finitely accessible category with products, and assume that its symmetric category [Formula: see text] is also finitely accessible and pure semisimple. We study necessary and sufficient conditions in both categories for [Formula: see text] (and hence [Formula: see text]) to be of locally finite representation type. In particular, we obtain a generalization of Herzog's criterion for finite representation type of left pure semisimple and right artinian rings. As an application, we prove that a left pure semisimple ring R with enough idempotents which has a self-duality is of locally finite representation type if and only if it is left locally finite.

Sign in / Sign up

Export Citation Format

Share Document