We say a group G is indecomposable if it cannot be written as a direct product of non-trivial subgroups G = H × K. If
Here, uniqueness means direct decompositions into indecomposable subgroups have the exchange property.
Moreover, any indecomposable subgroup is (trivially) the one-term direct product of itself, hence decomposable.
If Krull-Schmidt fails, then S contains G; so we may iteratively construct a descending series of direct factors; this contradicts the DCC.
[1] The proof of uniqueness, on the other hand, is quite long and requires a sequence of technical lemmas.
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
Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.
[4] In general, the theorem fails if one only assumes that the module is Noetherian or Artinian.
of Math (1909)), for finite groups, though he mentions some credit is due to an earlier study of G.A.
Wedderburn's theorem is stated as an exchange property between direct decompositions of maximum length.
The thesis of Robert Remak (1911) derived the same uniqueness result as Wedderburn but also proved (in modern terminology) that the group of central automorphisms acts transitively on the set of direct decompositions of maximum length of a finite group.
Otto Schmidt (Sur les produits directs, S. M. F. Bull.
41 (1913), 161–164), simplified the main theorems of Remak to the 3 page predecessor to today's textbook proofs.
His method improves Remak's use of idempotents to create the appropriate central automorphisms.
Both Remak and Schmidt published subsequent proofs and corollaries to their theorems.
Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions.
The name Krull–Schmidt is now popularly substituted for any theorem concerning uniqueness of direct products of maximum size.