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The Radical

In: Finite Dimensional Algebras

Author

Listed:
  • Yurij A. Drozd

    (Kiev Taras Shevchenko University, Faculty of Mechanics and Mathematics)

  • Vladimir V. Kirichenko

    (Kiev Taras Shevchenko University, Faculty of Mechanics and Mathematics)

Abstract

Theorems 2.4.3 (Wedderburn-Artin) and 2.6.2 give a complete description of semisimple algebras and their representations. In comparison, we know very little on the structure of non-semisimple algebras and modules over them, even in the case when K is algebraically closed. The fundamental concept here is the notion of a radical: the least ideal such that the respective quotient algebra is semisimple. An essential property of the radical is its nilpotency. It allows to “lift the idempotents modulo the radical”. In this way, the class of projective modules, related to semisimple modules, appears in a natural way. Their decomposition into the indecomposable ones can be shown to be unique, and by means of the endomorphism algebras, this result can be extended to arbitrary modules. Finally, in the last section of this chapter, we introduce the concept of a diagram of an algebra and of a universal algebra over a diagram; making use of them we obtain a description (of course, by no means complete) of algebras, at least in the algebraically closed case.5 In particular, we obtain the classification of so-called hereditary algebras (over an algebraically closed field).

Suggested Citation

  • Yurij A. Drozd & Vladimir V. Kirichenko, 1994. "The Radical," Springer Books, in: Finite Dimensional Algebras, chapter 3, pages 44-68, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-76244-4_3
    DOI: 10.1007/978-3-642-76244-4_3
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