Torsion (algebra) - Torsion in Homological Algebra

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 ring R (for example, two abelian groups, when R = Z), Tor functors yield a family of R-modules Tor_i(M,N). The S-torsion of an R-module M is canonically isomorphic to Tor₁(M, R_S/R). The symbol Tor 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.