Homomorphisms, Kernels and Prime Ideals

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Given an onto ring homomorphism Φ: R1 --> R2 we saw in class that if I C R1 is an ideal containing the kernel of Φ, then Φ(I) is an ideal of R2. In addition, we saw that if J C R2 is an ideal, then Φˉ¹(J) is an ideal of R1 containing kernel of Φ.

(i) Show that if I C R1 is a prime ideal containing the kernel of Φ, then Φ(I) is also prime.
(ii) Show that if J C R2 is prime, then Φˉ¹(J) is also prime.

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

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