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 of matrix rings over division rings.

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, then, for each, the linear endomorphism given by the natural projection followed by the natural inclusion is an idempotent. They are clearly orthogonal to each other and they sum up:
as endomorphisms. Conversely, each set of orthogonal idempotents such that only finitely many are nonzero for each and determine a direct sum decomposition by taking to be the images of.
This fact already puts some constraints on a possible decomposition of a ring: give a ring, suppose there is a decomposition
of as a left module over itself, where are left submodules; i.e., left ideals. Each endomorphism can be identified with a right multiplication by an element of R; thus, where are idempotents of. The summation of idempotent endomorphisms corresponds to the decomposition of the unity of R:, which is necessarily a finite sum; in particular, must be a finite set.
For example, take, the ring of n-by-n matrices over a division ring D. Then is the direct sum of n copies of, 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 decomposition of it as a left module over itself
into two-sided ideals of R. As above, for some orthogonal idempotents such that. Since is an ideal, and so for. Then, for each i,
That is, 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 itself is a ring on its own right, the unity given by, and, as a ring, R is the product ring
For example, again take. 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. 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 is said to complement maximal direct summands if for each maximal direct summand L of M, there exists a subset such that
Two decompositions are said to be equivalent if there is a bijection such that for each,. 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 such that the endomorphism ring of each is local, 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, then
  1. if nonzero, N contains an indecomposable direct summand,
  2. if is indecomposable, the endomorphism ring of it is local and is complemented by the given decomposition:
  3. : and so for some,
  4. for each, there exist direct summands of and of such that.
The endomorphism ring of an indecomposable module of finite length is local 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. Now, suppose we are given an indecomposable decomposition. Then it must be equivalent to the first one: so and for some permutation of. More precisely, since is indecomposable, for some. Then, since is indecomposable, and so on; i.e., complements to each sum can be taken to be direct sums of some 's.
Another application is the following statement :
To see this, choose a finite set such that. Then, writing, by Azumaya's theorem, with some direct summands of and then, by modular law, with. Then, since is a direct summand of, we can write and then, which implies, since F is finite, that 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 : is isomorphic to for some subset. According to, it is not known whether the assumption " 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:
  1. R is a semisimple ring; i.e., is a semisimple left module.
  2. where denotes the ring of n-by-n matrices and the positive integers are determined by R.
  3. Every left module over R is semisimple.
To see the equivalence of the first two, note: if where are mutually non-isomorphic left minimal ideals, then, with the view that endomorphisms act from the right,
where each can be viewed as the matrix ring over the division ring. The equivalence 1. 3. is because every module is a quotient of a free module and a quotient of a semisimple module is clearly semisimple.