Krull–Schmidt theorem

The set of Krull - Remak -Schmidt is an important sentence in algebra, a branch of mathematics. He states that can be written as a direct product of its irreducible subgroups or submodules clearly under certain finiteness conditions groups or modules essentially.

Set of Krull - Remak -Schmidt for Groups

Is a group that meets both the ascending and the descending chain condition for normal subgroups, it can be as a direct product of finitely many indecomposable subgroups of writing. Up to permutation and isomorphism indecomposable subgroups are uniquely determined.

Set of Krull - Remak -Schmidt for modules

If a module that is both noetherian and artinian, so it has finite length, so the direct sum of finitely many indecomposable modules is. Up to permutation and isomorphism, the indecomposable modules are uniquely determined.

History of the

In its present version, the set goes back to work of Robert Remak (1911 ), Wolfgang Krull (1925 ) and Otto Schmidt ( 1928).

The set of modules is generally wrong to only assumes that the module is artinian. This is the answer to a question that W. Krull had asked in 1932.

Swell

  • Thomas W. Hungerford: Algebra ( Graduate Texts in Mathematics, Vol 73). Springer, New York 2008, ISBN 0-387-90518-9 ( Nachdr d ed New York 1974).
  • Alberto Facchini: Modules story. Endomorphism rings and direct sum decompositions in some classes of modules (Progress in Mathematics, Vol 167). Birkhäuser, Basel 1998, ISBN 3-7643-5908-0.
  • Alberto Facchini, Dolores Herbera, Lawrence S. Levy, Peter Vamos: Krull -Schmidt fails for Artinian modules. In: Proceedings of the American Mathematical Society, Vol 123 (1995), Issue 12, pp. 3587-3592, ISSN 0002-9939.
  • Claus M. Ringel: Krull - Remak -Schmidt fails for Artinian modules over local rings. In: Algebras and Representation Theory, Vol 4 (2001 ), No. 1, pp. 77-86, ISSN 1386 - 923x.
  • Algebra
  • Set ( mathematics)
Noetherian ring Robert Remak (mathematician)
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