annihilator is an ideal

The right annihilator of a right R-module MR in R is an ideal.

By the distributive law for modules, it is easy to see that r.ann(MR) is closed under addition and right multiplication. Now take xr.ann(MR) and rR.

Take any mMR. Then mrMR, but then (mr)x=0 since xr.ann(MR). So m(rx)=0 and rxr.ann(MR).

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath result holds for left annihilators.

Title annihilatorMathworldPlanetmathPlanetmathPlanetmath is an ideal
Canonical name AnnihilatorIsAnIdeal
Date of creation 2013-03-22 12:50:27
Last modified on 2013-03-22 12:50:27
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Theorem
Classification msc 16D10
Classification msc 16D25