Case of A Principal Ideal Domain
Suppose that R is a (commutative) principal ideal domain and M is a finitely-generated R-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module M up to isomorphism. In particular, it claims that
where F is a free R-module of finite rank (depending only on M) and T(M) is the torsion submodule of M. As a corollary, any finitely-generated torsion-free module over R is free. This corollary does not hold for more general commutative domains, even for R = K, the ring of polynomials in two variables. For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a direct summand of it.
Read more about this topic: Torsion (algebra)
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