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I - Artin rings

Published online by Cambridge University Press:  11 May 2010

Maurice Auslander
Affiliation:
Brandeis University, Massachusetts
Idun Reiten
Affiliation:
Kunstakademiet i Trondheim, Norway
Sverre O. Smalo
Affiliation:
Kunstakademiet i Trondheim, Norway
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Summary

While we are assuming that the reader is familiar with general concepts of ring theory, such as the radical of a ring, and of module theory, such as projective, injective and simple modules, we are not assuming that the reader, except for semisimple modules and semisimple rings, is necessarily familiar with the special features of the structure of artin algebras and their finitely generated modules. This chapter is devoted to presenting background material valid for left artin rings, and the next chapter deals with special features of artin algebras. All rings considered in this book will be assumed to have an identity and all modules are unitary, and unless otherwise stated all modules are left modules.

We start with a discussion of finite length modules over arbitrary rings. After proving the Jordan–Hölder theorem, we introduce the notions of right minimal morphisms and left minimal morphisms and show their relationship to arbitrary morphisms between finite length modules. When applied to finitely generated modules over left artin rings, these results give the existence of projective covers which in turn gives the structure theorem for projective modules as well as the theory of idempotents in left artin rings. We also include some results from homological algebra which we will need in this book.

Finite length modules

In this section we introduce the composition series and composition factors for modules of finite length. We prove the Jordan–Hölder theorem and give an interpretation of it in terms of Grothendieck groups.

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Publisher: Cambridge University Press
Print publication year: 1995

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  • Artin rings
  • Maurice Auslander, Brandeis University, Massachusetts, Idun Reiten, Kunstakademiet i Trondheim, Norway, Sverre O. Smalo, Kunstakademiet i Trondheim, Norway
  • Book: Representation Theory of Artin Algebras
  • Online publication: 11 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511623608.002
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  • Artin rings
  • Maurice Auslander, Brandeis University, Massachusetts, Idun Reiten, Kunstakademiet i Trondheim, Norway, Sverre O. Smalo, Kunstakademiet i Trondheim, Norway
  • Book: Representation Theory of Artin Algebras
  • Online publication: 11 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511623608.002
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Artin rings
  • Maurice Auslander, Brandeis University, Massachusetts, Idun Reiten, Kunstakademiet i Trondheim, Norway, Sverre O. Smalo, Kunstakademiet i Trondheim, Norway
  • Book: Representation Theory of Artin Algebras
  • Online publication: 11 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511623608.002
Available formats
×