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A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free -modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free -module, and each free -module comes from a free abelian group in this way. [21]
This is the fundamental theorem of finitely generated abelian groups. The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums. [14]: 26–27
The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties .
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of A ( K ) {\displaystyle A(K)} to the zero of the associated L ...
An important step in the proof of the classification of finitely generated abelian groups is that every such torsion-free group is isomorphic to a . A non-finitely generated countable example is given by the additive group of the polynomial ring Z [ X ] {\displaystyle \mathbb {Z} [X]} (the free abelian group of countable rank).
Even if A is not finitely generated, the size of its torsion-free part is uniquely determined, as is explained in more detail in the article on rank of an abelian group. An abelian group A is torsion-free if and only if it is flat as a Z-module, which means that whenever C is a subgroup of some abelian group B, then the natural map from the ...
The Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian. [ 3 ] Jakob Nielsen ( 1921 ) originally proved a restricted form of the theorem, stating that any finitely-generated subgroup of a free group is free.