torsion element

Let R be a commutative ring, and M an R-module. We call an element mM a torsion element if there exists a non-zero-divisor αR such that αm=0. The set is denoted by tor(M).

tor(M) is not empty since 0tor(M). Let m,ntor(M), so there exist α,β0R such that 0=αm=βn. Since αβ(m-n)=βαm-αβn=0,αβ0, this implies that m-ntor(M). So tor(M) is a subgroupMathworldPlanetmathPlanetmath of M. Clearly τmtor(M) for any non-zero τR. This shows that tor(M) is a submodule of M, the torsion submodule of M. In particular, a module that equals its own torsion submodule is said to be a torsion module.

Title torsion element
Canonical name TorsionElement
Date of creation 2013-03-22 13:54:41
Last modified on 2013-03-22 13:54:41
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 13C12
Defines torsion submodule
Defines torsion module