Krull-Schmidt theorem

A group G is said to satisfy the ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath (or ACC) on normal subgroupsMathworldPlanetmath if there is no infiniteMathworldPlanetmath ascending proper chain G1G2G3 with each Gi a normal subgroup of G.

Similarly, G is said to satisfy the descending chain conditionMathworldPlanetmathPlanetmath (or DCC) on normal subgroups if there is no infinite descending proper chain of normal subgroups of G.

One can show that if a nontrivial group satisfies either the ACC or the DCC on normal subgroups, then that group can be expressed as the internal direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of finitely many indecomposableMathworldPlanetmath subgroupsMathworldPlanetmathPlanetmath. If both the ACC and DCC are satisfied, the Krull-Schmidt theorem guarantees that this “decomposition into indecomposables” is essentially unique. (Note that every finite groupMathworldPlanetmath satisfies both the ACC and DCC on normal subgroups.)

Krull-Schmidt theorem: Let G be a nontrivial group satisfying both the ACC and DCC on its normal subgroups. Suppose G=G1××Gn and G=H1××Hm (internal direct products) where each Gi and Hi is indecomposable. Then n=m and, after reindexing, GiHi for each i. Moreover, for all k<n, G=G1××Gk×Hk+1××Hn.

For proof, see Hungerford’s AlgebraMathworldPlanetmathPlanetmath.

NoetherianPlanetmathPlanetmath [resp. artinianPlanetmathPlanetmath] modules satisfy the ACC [resp. DCC] on submodulesMathworldPlanetmath. Indeed the Krull-Schmidt theorem also appears in the context of module theory. (Sometimes, as in Lang’s Algebra, this result is called the Krull-Remak-Schmidt theorem.)

Krull-Schmidt theorem (for modules): A nonzero module that is both noetherian and artinian can be expressed as the direct sumMathworldPlanetmathPlanetmath of finitely many indecomposable modules. These indecomposable summands are uniquely determined up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and permutationMathworldPlanetmath.


  • Hungerford, T., Algebra. New York: Springer, 1974.

  • Lang, S., Algebra. (3d ed.), New York: Springer, 2002.

Title Krull-Schmidt theorem
Canonical name KrullSchmidtTheorem
Date of creation 2013-03-22 15:24:00
Last modified on 2013-03-22 15:24:00
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 24
Author CWoo (3771)
Entry type Theorem
Classification msc 16P40
Classification msc 16P20
Classification msc 16D70
Classification msc 20E34
Classification msc 20-00
Synonym Krull-Remak-Schmidt theorem
Related topic IndecomposableGroup
Defines ascending chain condition
Defines descending chain condition