Mathematics Homework Solutions
#37877
Homomorphisms, Kernels and Prime Ideals
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.
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.
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