# Ideals and Factor Rings : Annihilators

If R is a commutative ring, a polynomial f(x) in R[x] is said to annihilate R if f(a) = 0 for every a belonging to R

Show that x^p - x annihilates Zp (Z is integers).

Â© BrainMass Inc. brainmass.com March 4, 2021, 5:54 pm ad1c9bdddfhttps://brainmass.com/math/ring-theory/ideals-factor-rings-annihilators-fermats-theorem-17293

#### Solution Preview

Proof:

By Fermat's theorem, if p is a prime, then x^(p-1)=1 (mod p) for any x in Zp. So ...

#### Solution Summary

An annihilator is identified and Fermat's theorem is used.

$2.19