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Math Help - Prime Ideals of Commutative Rings

  1. #1
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    Prime Ideals of Commutative Rings

    Let phi: R --> R' be a homomorphism where both R and R' are commutative rings and I is a subring of R. Show that if J is a prime ideal of R' and I = phi^-1(J), then I is a prime ideal of R.
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    Quote Originally Posted by Coda202 View Post

    Let \phi: R --> R' be a homomorphism where both R and R' are commutative rings and I is a subring of R. Show that if J is a prime ideal of R' and I = \phi^{-1}(J), then I is a prime ideal of R.
    I is an ideal of R because J is an ideal of R' and I=\phi^{-1}(J)=\{a \in R: \ \phi(a) \in J \}. we don't need any assumptions on I (including assuming that I is a subring!!).

    to show that I is prime, just follow the definition again: if ab \in I, then \phi(a) \phi(b)=\phi(ab) \in J. but J is a prime ideal of R'. so either \phi(a) \in J or \phi(b) \in J. hence either a \in I or b \in I.
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