Krull-Remak-Schmidt theorem

The Krull-Remak-Schmidt theorem is an important theorem in algebra , a branch of mathematics . It states that, under certain finiteness requirements, groups or modules can essentially be clearly written as a direct product of their indivisible subgroups or sub- modules .

Krull-Remak-Schmidt theorem for groups

If a group fulfills both the ascending and the descending chain condition for normal subgroups, then the direct product of a finite number of indivisible subgroups can be written. Except for permutation and isomorphism, the indivisible subgroups are clearly defined. ${\ displaystyle G}$ ${\ displaystyle G}$ ${\ displaystyle G}$

Krull-Remak-Schmidt theorem for modules

If a module is both Noetherian and Artinian, i.e. has finite length , then the direct sum of finitely many indivisible modules is. Except for permutation and isomorphism, the indivisible modules are clearly defined. ${\ displaystyle M \ neq 0}$ ${\ displaystyle M}$

History of the sentence

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

The theorem for modules is generally wrong, assuming only that the module is Artinian. That is the answer to a question that W. Krull asked in 1932.

swell

Thomas W. Hungerford: Algebra (Graduate Texts in Mathematics; Vol. 73). Springer, New York 2008, ISBN 0-387-90518-9 (reprint of the New York 1974 edition).
Alberto Facchini: Module 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 Vámos: Krull-Schmidt fails for Artinian modules. In: Proceedings of the American Mathematical Society , Vol. 123 (1995), No. 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), Issue 1, pp. 77-86, ISSN 1386-923X .