Krull–Schmidt theorem

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In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

Contents 1 Definitions 2 Krull–Schmidt theorem 3 Krull–Schmidt theorem for modules 4 History 5 See also 6 Further reading 7 External links

[edit] Definitions

We say that a group G satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of G:

is eventually constant, i.e., there exists N such that G_N = G_N+1 = G_N+2 = ... . We say that G satisfies the ACC on normal subgroups if every such sequence of normal subgroups of G eventually becomes constant.

Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups:

Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group satisfies ACC but not DCC, since (2) > (2)² = (2)³ > ... is an infinite decreasing sequence of subgroups. On the other hand, the -torsion part of (the quasicyclic p-group) satisfies DCC but not ACC.

We say a group G is indecomposable if it cannot be written as a direct product of non-trivial subgroups G = H × K.

[edit] Krull–Schmidt theorem

The theorem says:

If G is a group that satisfies ACC and DCC on normal subgroups, then there is a unique way of writing G as a direct product of finitely many indecomposable subgroups of G. Here, uniqueness means: suppose is another expression of G as a product of indecomposable subgroups. Then k = l and there is a reindexing of the H_i's satisfying

• G_i and H_i are isomorphic for each i;
• for each r.

[edit] Krull–Schmidt theorem for modules

If is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian or – equivalently – of finite length), then E is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.

In general, the theorem fails, if one only assumes that the module is Artinian.

[edit] History

The present-day Krull–Schmidt theorem is the result of work by Robert Remak (1911), Wolfgang Krull (1925) and Otto Schmidt (1928) in a paper Über unendliche Gruppen mit endlicher Kette.

[edit] See also

• Krull–Schmidt category

[edit] Further reading

• Hungerford, Thomas W. Algebra, Graduate Texts in Mathematics Volume 73. ISBN 0-387-90518-9
• A. Facchini: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress in Mathematics, 167. Birkhäuser Verlag, Basel, 1998. ISBN 3-7643-5908-0
• A. Facchini, D. Herbera, L.S. Levy, P. Vámos: Krull-Schmidt fails for Artinian modules. Proc. Amer. Math. Soc. 123 (1995), no. 12, 3587–3592.
• C.M. Ringel: Krull-Remak-Schmidt fails for Artinian modules over local rings. Algebr. Represent. Theory 4 (2001), no. 1, 77–86.

[edit] External links

• Page at PlanetMath