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    Homomorphisms, Kernels and Prime Ideals

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    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.

    © BrainMass Inc. brainmass.com October 9, 2019, 4:38 pm ad1c9bdddf
    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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