# Homomorphisms, Kernels and Prime Ideals

Given an onto ring homomorphism Φ: R1 --> R2 we saw in class that if I <u>C</u> R1 is an ideal containing the kernel of Φ, then Φ(I) is an ideal of R2. In addition, we saw that if J <u>C</u> R2 is an ideal, then ΦˉÂ¹(J) is an ideal of R1 containing kernel of Φ.

(i) Show that if I <u>C</u> R1 is a prime ideal containing the kernel of Φ, then Φ(I) is also prime.

(ii) Show that if J <u>C</u> R2 is prime, then ΦˉÂ¹(J) is also prime.

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

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Proof:

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

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