Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory, is known as the Krull–Schmidt theorem.
A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a product) of matrix rings over division rings (this observation is known as the Artin–Wedderburn theorem).
To give a direct sum decomposition of a module into submodules is the same as to give orthogonal idempotents in the endomorphism ring of the module that sum up to the identity map.
) and they sum up to the identity map: as endomorphisms (here the summation is well-defined since it is a finite sum at each element of the module).
Conversely, each set of orthogonal idempotents
determine a direct sum decomposition by taking
[2] The summation of idempotent endomorphisms corresponds to the decomposition of the unity of R:
, the columns; each column is a simple left R-submodule or, in other words, a minimal left ideal.
Suppose there is a (necessarily finite) decomposition of it as a left module over itself into two-sided ideals
[4] Clearly, the argument can be reversed and so there is a one-to-one correspondence between the direct sum decomposition into ideals and the orthogonal central idempotents summing up to the unity 1.
There are several types of direct sum decompositions that have been studied: Since a simple module is indecomposable, a semisimple decomposition is an indecomposable decomposition (but not conversely).
If the endomorphism ring of a module is local, then, in particular, it cannot have a nontrivial idempotent: the module is indecomposable.
A direct summand is said to be maximal if it admits an indecomposable complement.
is said to complement maximal direct summands if for each maximal direct summand L of M, there exists a subset
[7] If a module admits an indecomposable decomposition complementing maximal direct summands, then any two indecomposable decompositions of the module are equivalent.
[8] In the simplest form, Azumaya's theorem states:[9] given a decomposition
The more precise version of the theorem states:[10] still given such a decomposition, if
, then The endomorphism ring of an indecomposable module of finite length is local (e.g., by Fitting's lemma) and thus Azumaya's theorem applies to the setup of the Krull–Schmidt theorem.
Indeed, if M is a module of finite length, then, by induction on length, it has a finite indecomposable decomposition
, which is a decomposition with local endomorphism rings.
Another application is the following statement (which is a key step in the proof of Kaplansky's theorem on projective modules): To see this, choose a finite set
for some J by a repeated application of Azumaya's theorem.
In the setup of Azumaya's theorem, if, in addition, each
is countably generated, then there is the following refinement (due originally to Crawley–Jónsson and later to Warfield):
[12] (In a sense, this is an extension of Kaplansky's theorem and is proved by the two lemmas used in the proof of the theorem.)
countably generated" can be dropped; i.e., this refined version is true in general.
On the decomposition of a ring, the most basic but still important observation, known as the Wedderburn-Artin theorem is this: given a ring R, the following are equivalent: To show 1.
are mutually non-isomorphic minimal left ideals.
, which is a division ring by Schur's Lemma.
The converse holds because the decomposition of 2. is equivalent to a decomposition into minimal left ideals = simple left submodules.