Editing Semisimple module (section)

== Definition ==
A [[module (mathematics)|module]] over a (not necessarily commutative) ring is said to be '''semisimple''' (or '''completely reducible''') if it is the [[direct sum of modules|direct sum]] of [[simple module|simple]] (irreducible) submodules.

For a module ''M'', the following are equivalent:
# ''M'' is semisimple; i.e., a direct sum of irreducible modules.
# ''M'' is the sum of its irreducible submodules.
# Every submodule of ''M'' is a [[direct summand]]: for every submodule ''N'' of ''M'', there is a complement ''P'' such that {{nowrap|1=''M'' = ''N'' ⊕ ''P''}}.
For the proof of the equivalences, see ''{{section link|Semisimple representation#Equivalent characterizations}}''.<!--
For <math>3 \Rightarrow 2</math>, the starting idea is to find an irreducible submodule by picking any nonzero <math>x\in M</math> and letting <math>P</math> be a [[maximal submodule]] such that <math>x \notin P</math>.-- by Zorn's lemma? -- It can be shown that the complement of <math>P</math> is irreducible.{{sfn|ps=|Jacobson|1989|p=120}} -->

The most basic example of a semisimple module is a module over a field, i.e., a [[vector space]]. On the other hand, the ring {{nowrap|'''Z'''}} of integers is not a semisimple module over itself, since the submodule {{nowrap|2'''Z'''}} is not a direct summand.

Semisimple is stronger than [[indecomposable module|completely decomposable]],
which is a [[direct sum of modules|direct sum]] of [[indecomposable module|indecomposable submodules]].

Let ''A'' be an algebra over a field ''K''. Then a left module ''M'' over ''A'' is said to be '''absolutely semisimple''' if, for any field extension ''F'' of ''K'', {{nowrap|''F'' ⊗<sub>''K''</sub> ''M''}} is a semisimple module over {{nowrap|''F'' ⊗<sub>''K''</sub> ''A''}}.