Torsion (algebra) - Torsion and Localization

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 M_Q, 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 R_S. There is a canonical map from M to M_S, 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 Ore condition, or more generally for any right denominator set S and right R-module M.