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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 $G_{1}\subsetneq G_{2}\subsetneq G_{3}\cdots$ 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 $G=G_{1}\times\cdots\times G_{n}$ and $G=H_{1}\times\cdots\times H_{m}$ (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 $k<n$, $G=G_{1}\times\cdots\times G_{k}\times H_{{k+1}}\times\cdots\times H_{n}$.
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.

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

Lang, S., Algebra. (3d ed.), New York: Springer, 2002.
Mathematics Subject Classification
16P40 no label found16P20 no label found16D70 no label found20E34 no label found2000 no label found Forums
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