|
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. == Definition == A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of 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 ''M'' = ''N'' ⊕ ''P''. For , the starting idea is to find an irreducible submodule by picking any nonzero and letting be a maximal submodule such that . It can be shown that the complement of is irreducible.〔Nathan Jacobson, Basic Algebra II (Second Edition), 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 Z of integers is not a semisimple module over itself (because, for example, it is not an artinian ring.) Semisimple is stronger than completely decomposable, which is a direct sum of 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'', is a semisimple module over . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「semisimple module」の詳細全文を読む スポンサード リンク
|