Remarks on a paper by Skornyakov concerning rings for which every module is a direct sum of left ideals

Jøndrup, S.; Ringel, Claus Michael

doi:10.1007/bf01226456

Cited by 5 publications

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“…The quiver approach has lately been used to classify many of the quasi-Frobenius algebras of finite representation type; and Jondrup and Ringel [149]. Further, it can be utilized for integral…”

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The history of algebras and their representations

Gustafson¹

1982

Lecture Notes in Mathematics

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In this paper, I have attempted to outline the origins of the theory of finite dimensional algebras over fields, with particular attention to representation theory.Hence, the treatment is not exhaustive; for instance, we say nothing about the extensive work done on the structure of division algebras and little about group representations per se. Even with such provisos, much had to be omitted, due to the extensiveness of the literature (see for example the bibliography [276], which lists more than seven hundred related items for the years 1969-1979 alone). I have therefore chosen to consider what I myself feel are the main lines of development, realizing that many disagree with my choices. In order to maintain an upper bound on the length of this work, I have omitted familiar definitions, and provided no mathematical exposition at all. Hence, the beginner who stumbles upon this paper and wishes to learn more will have to go to the cited references. Good starting points for learning the classical theory are the text books of Albert [i] , Artin, Nesbitt and Thrall [5], Curtis and Reiner [62], Deuring [67], Dickson [71,72,74] and Jacobson [143]. Unfortunately, no text is available at the time of this writing that covers the powerful methods developed in the last few years, but some guidance can be found in the expository papers of Gabriel [106,107,108]. Something of the approach of M. Auslander can be learned from the notes [6,14]. As I have said, the scope of this paper is strictly limited and the format is condensed --it is more a chronology or an annotated bibliography than a scholarly history. I highly recommend that one also examine other sources, such as the excellent papers of Hawkins [132, 133,134], which are also summarized in Mackey [183], as well as Artin [4], Happel [129], Ringel [233] and Wussing [270]. I have used these sources freely in preparing this report, especially the works of Hawkins and Ringel. It is not unreasonable to think that the theory of algebras begins in 1835, when William Rowan Hamilton represented complex numbers as ordered pairs of real numbers (his paper [125] on this subject appeared in 1837). This had been done before, in a sense, in the geometric representation of complex numbers. However, Hamilton was the first to see the algebraic significance of such a representation: it reflectsthe fact that the plus-sign in a+bi has a fundamentally different meaning than that which it has in the arithmetic of real numbers.

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confidence: 99%

The history of algebras and their representations

Gustafson¹

1982

Lecture Notes in Mathematics

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Modules

Markov¹,

Mikhalëv²,

Skornyakov³

et al. 1983

J Math Sci

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Kawada's theorem

Ringel¹

1981

Lecture Notes in Mathematics

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Kawadas's theorem solved the KSthe problem for basic finite-dimensional algebras: It characterizes completely those finite-dimensional algebras for which any indecomposable module has squarefree socle and squarefree top, and describes the possible indecomposable modules. This seems to be the most elaborate result of the classical representation theory (prior to the introduction of the new combinatorical and homological tools: quivers, partially ordered sets, vectorspace categories, Auslander-Reiten sequences). However, apparently his work was not appreciated at that time. These are the revised notes of parts of a series of lectures given at the meeting on abelian groups and modules in Trento (Italy), 1980. They are centered around the second part of Kawada's theorem: the shapes of the indecomposable modules over a Kawada algebra. I. K~the algebras and algebras of finite representation type Recall the following important property of abelian groups, thus of ~-modules: every finit~ygenerated module is a direct sum of cyclic modules. KSthe showed that the only commutative finite-dimensional algebras which have this property are the uniserial ones, and he posed the question to classify also the non-commutative finitedimensional algebras with this property [II]. An algebra for which any finitely generated left or right module is a direct sum of cyclic modules, is now called a KSthe-algebra, and a classification of these algebras seems to be rather difficult.

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Remarks on a paper by Skornyakov concerning rings for which every module is a direct sum of left ideals

Cited by 5 publications

References 4 publications

The history of algebras and their representations

The history of algebras and their representations

Modules

Kawada's theorem

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