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structure theorem for finitely generated modules over a principal ideal domain
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
==Statement==
When a vector space over a field ''F'' has a finite generating set, then one may extract from it a basis consisting of a finite number ''n'' of vectors, and the space is therefore isomorphic to ''Fn''. The corresponding statement with the ''F'' generalized to a principal ideal domain ''R'' is no longer true, as a finitely generated module over ''R'' need not have any basis. However such a module is still isomorphic to a quotient of some module ''Rn'' with ''n'' finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis ''Rn'' to the generators of the module, and take the quotient by its kernel.) By changing the choice of generating set, one can in fact describe the module as the quotient of some ''Rn'' by a particularly simple submodule, and this is the structure theorem.
The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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