annihilator


Let R be a ring, and suppose that M is a left R-module and N a right R-module.

Annihilator of a Subset of a Module

  1. 1.

    If X is a subset of M, then we define the left annihilator of X in R:

    l.ann(X)={rRrx=0 for all xX}.

    If a,bl.ann(X), then so are a-b and ra for all rR. Therefore, l.ann(X) is a left idealMathworldPlanetmathPlanetmath of R.

  2. 2.

    If Y is a subset of N, then we define the right annihilator of Y in R:

    r.ann(Y)={rRyr=0 for all yY}.

    Like above, it is easy to see that r.ann(Y) is a right ideal of R.

Remark. l.ann(X) and r.ann(Y) may also be written as l.annR(X) and r.annR(Y) respectively, if we want to emphasize R.

Annihilator of a Subset of a Ring

  1. 1.

    If Z is a subset of R, then we define the right annihilator of Z in M:

    r.annM(Z)={mMzm=0 for all zZ}.

    If m,nr.annM(Z), then so are m-n and rm for all rR. Therefore, r.annM(Z) is a left R-submodule of M.

  2. 2.

    If Z is a subset of R, then we define the left annihilator of Z in N:

    l.annN(Z)={nNnz=0 for all zZ}.

    Similarly, it can be easily seen that l.annN(Z) is a right R-submodule of N.

Title annihilatorPlanetmathPlanetmathPlanetmath
Canonical name Annihilator
Date of creation 2013-03-22 12:01:32
Last modified on 2013-03-22 12:01:32
Owner antizeus (11)
Last modified by antizeus (11)
Numerical id 8
Author antizeus (11)
Entry type Definition
Classification msc 16D10
Synonym left annihilator
Synonym right annihilator
Related topic JacobsonRadical