Here, uniqueness means direct decompositions into indecomposable subgroups have the exchange property.
2.
Wedderburn's theorem is stated as an exchange property between direct decompositions of maximum length.
3.
Some authors choose to call direct decompositions of maximum-size Remak decompositions to honor his contributions.
4.
Just as idempotents determine the direct decompositions of " R " as a module, the central idempotents of " R " determine the decompositions of " R " as a direct sum of rings.
5.
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.
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