KrullSchmidt theorem
A group $G$ is said to satisfy the ascending chain condition^{} (or ACC) on normal subgroups^{} if there is no infinite^{} ascending proper chain ${G}_{1}\u228a{G}_{2}\u228a{G}_{3}\mathrm{\cdots}$ with each ${G}_{i}$ a normal subgroup of $G$.
Similarly, $G$ is said to satisfy the descending chain condition^{} (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 product^{} of finitely many indecomposable^{} subgroups^{}. If both the ACC and DCC are satisfied, the KrullSchmidt theorem guarantees that this “decomposition into indecomposables” is essentially unique. (Note that every finite group^{} satisfies both the ACC and DCC on normal subgroups.)
KrullSchmidt theorem: Let $G$ be a nontrivial group satisfying both the ACC and DCC on its normal subgroups. Suppose $G={G}_{1}\times \mathrm{\cdots}\times {G}_{n}$ and $G={H}_{1}\times \mathrm{\cdots}\times {H}_{m}$ (internal direct products) where each ${G}_{i}$ and ${H}_{i}$ is indecomposable. Then $n=m$ and, after reindexing, ${G}_{i}\cong {H}_{i}$ for each $i$. Moreover, for all $$, $G={G}_{1}\times \mathrm{\cdots}\times {G}_{k}\times {H}_{k+1}\times \mathrm{\cdots}\times {H}_{n}$.
For proof, see Hungerford’s Algebra^{}.
Noetherian^{} [resp. artinian^{}] modules satisfy the ACC [resp. DCC] on submodules^{}. Indeed the KrullSchmidt theorem also appears in the context of module theory. (Sometimes, as in Lang’s Algebra, this result is called the KrullRemakSchmidt theorem.)
KrullSchmidt theorem (for modules): A nonzero module that is both noetherian and artinian can be expressed as the direct sum^{} of finitely many indecomposable modules. These indecomposable summands are uniquely determined up to isomorphism^{} and permutation^{}.
References.

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

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Lang, S., Algebra. (3d ed.), New York: Springer, 2002.
Title  KrullSchmidt theorem 
Canonical name  KrullSchmidtTheorem 
Date of creation  20130322 15:24:00 
Last modified on  20130322 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 2000 
Synonym  KrullRemakSchmidt theorem 
Related topic  IndecomposableGroup 
Defines  ascending chain condition 
Defines  descending chain condition 