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Homomorphisms, Kernels and Prime Ideals

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.

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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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