Decomposition of a module

In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple if and only if every module over it is a semisimple module.
An indecomposable module is a module that is not a direct sum of two nonzero submodules. 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).
Idempotents and decompositions.
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. Indeed, if formula_1, then, for each formula_2, the linear endomorphism formula_3 given by the natural projection followed by the natural inclusion is an idempotent. They are clearly orthogonal to each other (formula_4 for formula_5) and they sum up:
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 formula_7 such that only finitely many formula_8 are nonzero for each formula_9 and formula_10 determine a direct sum decomposition by taking formula_11 to be the images of formula_12.
This fact already puts some constraints on a possible decomposition of a ring: give a ring formula_13, suppose there is a decomposition
of formula_13 as a left module over itself, where formula_16 are left submodules; i.e., left ideals. Each endomorphism formula_17 can be identified with a right multiplication by an element of "R"; thus, formula_18 where formula_19 are idempotents of formula_20. The summation of idempotent endomorphisms corresponds to the decomposition of the unity of "R": formula_21, which is necessarily a finite sum; in particular, formula_22 must be a finite set.
For example, take formula_23, the ring of "n"-by-"n" matrices over a division ring "D". Then formula_24 is the direct sum of "n" copies of formula_25, the columns; each column is a simple left "R"-submodule or, in other words, a minimal left ideal.
Let "R" be a ring. Suppose there is a (necessarily finite) decomposition of it as a left module over itself
into "two-sided ideals" formula_27 of "R". As above, formula_28 for some orthogonal idempotents formula_12 such that formula_30. Since formula_27 is an ideal, formula_32 and so formula_33 for formula_5. Then, for each "i",
That is, formula_12 are in the center; i.e., they are central idempotents. 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. Also, each formula_27 itself is a ring on its own right, the unity given by formula_12, and, as a ring, "R" is the product ring formula_39
For example, again take formula_23. This ring is a simple ring; in particular, it has no nontrivial decomposition into two-sided ideals.
Types of decomposition.
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. Thus, a decomposition with local endomorphism rings is an indecomposable decomposition.
A direct summand is said to be "maximal" if it admits an indecomposable complement. A decomposition formula_41 is said to "complement maximal direct summands" if for each maximal direct summand "L" of "M", there exists a subset formula_42 such that 
Two decompositions formula_44 are said to be "equivalent" if there is a bijection formula_45 such that for each formula_2, formula_47. If a module admits an indecomposable decomposition complementing maximal direct summands, then any two indecomposable decompositions of the module are equivalent.
Azumaya's theorem.
In the simplest form, Azumaya's theorem states: given a decomposition formula_41 such that the endomorphism ring of each formula_11 is local (so the decomposition is indecomposable), each indecomposable decomposition of "M" is equivalent to this given decomposition. The more precise version of the theorem states: still given such a decomposition, if formula_50, 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 formula_62, which is a decomposition with local endomorphism rings. Now, suppose we are given an indecomposable decomposition formula_63. Then it must be equivalent to the first one: so formula_64 and formula_65 for some permutation formula_66 of formula_67. More precisely, since formula_68 is indecomposable, formula_69 for some formula_70. Then, since formula_71 is indecomposable, formula_72 and so on; i.e., complements to each sum formula_73 can be taken to be direct sums of some formula_11's.
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 formula_81 such that formula_82. Then, writing formula_83, by Azumaya's theorem, formula_84 with some direct summands formula_85 of formula_86 and then, by modular law, formula_87 with formula_88. Then, since formula_89 is a direct summand of formula_90, we can write formula_91 and then formula_92, which implies, since "F" is finite, that formula_93 for some "J" by a repeated application of Azumaya's theorem.
In the setup of Azumaya's theorem, if, in addition, each formula_11 is countably generated, then there is the following refinement (due originally to Crawley–Jónsson and later to Warfield): formula_51 is isomorphic to formula_96 for some subset formula_42. (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.) According to , it is not known whether the assumption "formula_11 countably generated" can be dropped; i.e., this refined version is true in general.
Decomposition of a ring.
On the decomposition of a ring, the most basic but still important observation, known as the Artin–Wedderburn theorem is this: given a ring "R", the following are equivalent:
To see the equivalence of the first two, note: if formula_104 where formula_105 are mutually non-isomorphic left minimal ideals, then, with the view that endomorphisms act from the right,
where each formula_107 can be viewed as the matrix ring over the division ring formula_108. (The converse is because the decomposition of 2. is equivalent to a decomposition into minimal left ideals = simple left submodules.) The equivalence 1. formula_109 3. is because every module is a quotient of a free module and a quotient of a semisimple module is clearly semisimple.