be generalized as follows: if R is an integral domain and Q is its field of fractions, then Q/R is a torsion R-module. The torsion subgroup of (R/Z, +) is (Q/Z, +) while the groups (R, +) and (Z, +) are torsion-free. The quotient of a torsion-free abelian group by a subgroup is torsion-free exactly when the subgroup is a pure subgroup. Consider a linear operator L acting on a finite-dimensional vector space V. If we view V as an F[L]-module in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the Cayley–Hamilton theorem), V is a torsion F[L]-module. 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[x,y], 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. Torsion and localization Assume that R is a commutative domain and M is an R-module. Let Q be the quotient field of the ring R. Then one can consider the Q-module obtained from M by extension of scalars. Since Q is a field, a module over Q is a vector space, possibly infinite-dimensional. There is a canonical homomorphism of abelian groups from M to MQ, and the kernel of this homomorphism is precisely the torsion submodule T(M). More generally, if S is a multiplicatively closed subset of the ring R, then we may consider localization of the R-module M, which is a module over the localization RS. There is a canonical map from M to MS, whose kernel is precisely the S-torsion submodule of M. Thus the torsion submodule of M can be interpreted as the set of the elements that "vanish in the localization". The same interpretation continues to hold in the non-commutative setting for rings satisfying the