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Math Help - Ring homomorphism

  1. #1
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    Ring homomorphism

    Let alpha : R ---> S be a ring homomorphism

    (i) If R is a principal ideal domain (PID) and S is an integral domain. Show that alpha is EITHER injective OR Im(alpha) is a field.

    (ii) Let S be a integral domain. By considering a ring homomorphism e_0: R = S[x]----> S by f(x) ---> f(0). Show that if R=S[x] is a PID, then S is a field.

    Ok, in part (i), we have to prove that Im(alpha) is field if and only if Ker(alpha) is a maximal ideal of R. But I don't know how to prove it.

    Part (ii) looks a bit hard to me, I dont know where to start

    Any ideas would be greatful

    Thank you so much
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  2. #2
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    we have to prove that Im(alpha) is field if and only if Ker(alpha) is a maximal ideal of R. But I don't know how to prove it.
    You may have in your course something (first isomorphism theorem?) like: if f is a ring morphism between two rings A and B, then f(A)\cong A/\text{ker}f
    The fact that for any ring R, the quotient of R by one of its ideal I is a field iff I is a maximal ideal will allow you to conclude.

    ii) So a way to prove that S is a field is to show that e_0 is surjective homomorphism (i.e. e_0(R)=S) and that \text{ker}e_0 is a maximal ideal of R.

    Therefore the first question is : what is \text{ker}e_0 ?
    Last edited by clic-clac; November 15th 2009 at 09:27 AM. Reason: name
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