# Krull-Schmidt 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 Krull-Schmidt theorem guarantees that this “decomposition into indecomposables” is essentially unique. (Note that every finite group 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 $G=G_{1}\times\cdots\times G_{n}$ 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, $G=G_{1}\times\cdots\times G_{k}\times H_{k+1}\times\cdots\times H_{n}$.

For proof, see Hungerford’s Algebra.

Noetherian [resp. artinian] modules satisfy the ACC [resp. DCC] on submodules. Indeed the 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 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.

 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