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Concept
Decomposition of a module
Summary
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).
Idempotent element
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 ( for ) 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 such that only finitely many are nonzero for each and determine a direct sum decomposition by taking to be the of .
This fact already puts some constraints on a possible decomposition of a ring: given a ring , suppose there is a decomposition
of as a left module over itself, where are left submodules; i.e., left ideals.
Official source
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