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Math Help - Fews rings question

  1. #1
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    Fews rings question

    I have few questions about rings concept that i not really sure:

    (a)Let F be field. Every ideal in F[x] is a prime ideal.

    (b)Let F be field. Every ideal in F[x] is a principal ideal.(I got this one)

    (c)If \delta is Euclidean norm on Euclidean domain D then \delta(a)=\delta(b) if a,b \in D are associates.


    Please let me know is that correct, little bit of explain is ok. I'm not looking for a proof. Thank you
    Last edited by kleenex; June 5th 2008 at 06:44 PM.
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  2. #2
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    Quote Originally Posted by kleenex View Post
    (a)Let F be field. Every ideal in F[x] is a prime ideal.
    this is very false. for example, <x^2> is not prime.

    (b)Let F be field. Every ideal in F[x] is a principal ideal. (I got this one)
    good! just note that in general every Euclidean domain is PID.

    (c)If \delta is Euclidean norm on Euclidean domain D then \delta(a)=\delta(b) if a,b \in D are associates.
    recall that \delta(x) \leq \delta(xy) for all nonzero elements x, y in D. now if a, b are associates,

    then a=ub, for some unit u \in D. so \delta(a) \leq \delta(u^{-1}a) = \delta(b) \leq \delta(ub)=\delta(a). \ \ \square
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