Ore condition, or more generally for any right denominator set S and right R-module M. Torsion in homological algebra The concept of torsion plays an important role in homological algebra. If M and N are two modules over a commutative domain R (for example, two abelian groups, when R = Z), Tor functors yield a family of R-modules Tori (M,N). The S-torsion of an R-module M is canonically isomorphic to TorR1(M, RS/R) by the exact sequence of TorR*: The short exact sequence of R-modules yields an exact sequence , hence is the kernel of the localisation map of M. The symbol denoting the functors reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set S is a right denominator set. Abelian varieties The torsion elements of an abelian variety are torsion points or, in an older terminology, division points. On elliptic curves they may be computed in terms of division polynomials. See also Analytic torsion Arithmetic dynamics Flat module Annihilator (ring theory) Localization of a module Rank of an abelian group Ray–Singer torsion Torsion-free abelian group Universal coefficient theorem References Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, Irving Kaplansky, "Infinite abelian groups", University of Michigan, 1954. Abelian group theory Module theory Homological algebra