Explore BrainMass

# Homomorphisms, Kernels and Prime Ideals

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

Given an onto ring homomorphism &#934;: R1 --> R2 we saw in class that if I <u>C</u> R1 is an ideal containing the kernel of &#934;, then &#934;(I) is an ideal of R2. In addition, we saw that if J <u>C</u> R2 is an ideal, then &#934;&#713;¹(J) is an ideal of R1 containing kernel of &#934;.

(i) Show that if I <u>C</u> R1 is a prime ideal containing the kernel of &#934;, then &#934;(I) is also prime.
(ii) Show that if J <u>C</u> R2 is prime, then &#934;&#713;¹(J) is also prime.

https://brainmass.com/math/linear-transformation/homomorphisms-kernels-prime-ideals-37877

#### Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

Proof:
1. If is a prime ideal containing , we want to show ...

#### Solution Summary

Homomorphisms, Kernels and Prime Ideals are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

\$2.19