# Ideals and Factor Rings : Annihilator

Problem:

If X is contained in R is a nonempty subset of a commutative ring R, define the annihilator of X by ann(X) = { a belonging to R | ax=0 for all x belonging to X}

Show that X is contained in ann[ann(X)]

AND

Show that ann(X) = ann{ann[ann(X)]}.

https://brainmass.com/math/ring-theory/ideals-factor-rings-annihilator-17285

#### Solution Preview

Proof:

X is contained in R and R is commutative, so we have ax=xa for any elements a,x in R.

First, I show that X is contained in ann[ann(X)]

By definition of ann(X), we know, for any x in X and any y in ann(X), we have yz=0 ...

#### Solution Summary

A proof involving an annihilator is provided and the details are in the solution.

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