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Thread: 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 $\displaystyle \phi$: 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 = $\displaystyle \phi^{-1}$(J), then I is a prime ideal of R.
    $\displaystyle I$ is an ideal of $\displaystyle R$ because $\displaystyle J$ is an ideal of $\displaystyle R'$ and $\displaystyle I=\phi^{-1}(J)=\{a \in R: \ \phi(a) \in J \}.$ we don't need any assumptions on $\displaystyle I$ (including assuming that $\displaystyle I$ is a subring!!).

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