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    • Thread Starter

    I am wondering whether or not my proof is correct (Please correct me if I am wrong). I must prove the following:

    "Suppose that R and S are isomorphic rings. Prove that R is an integral domain if and only if S is an integral domain."

    PROOF: (I know that there are two parts to the proof because it's an 'if and only if' proof. This half of the proof aims to show that S is an I.D iff R is an I.D. Hopefully if I can correct this part of the proof the other half will be easy to correct)

    Let f,g be elements of S-{0}, where f=Ø(a) for some a in R-{0} and g=Ø(b) for some b in R-{0}.

    Since S is an I.D., f.g = g.f ≠ 0 for all f,g in S-{0}.

    Hence, Ø(a).Ø(b) ≠ 0 so Ø(ab) ≠ 0

    and Ø(b).Ø(a) ≠ 0 so Ø(ba) ≠ 0.

    So for all a,b in R-{0}, ab ≠ 0. Hence R has no zero divisors.

    We know that S is commutative,
    hence Ø(a).Ø(b) = Ø(ab) = Ø(b)Ø(a) = Ø(ba)
    and therefore R is commutative.

    S has a 1 ≠ 0 .
    So, Ø(a).1 = Ø(a).Ø(1) = Ø(a.1).
    So R has a 1 ≠ 0.

    Hence, S is an I.D iff R is an I.D.
    Thank you for your time!

    Looks OK so far.

    Doing the "if" implies the "only if" as the inverse of the isomorphism will also be an isomorphism.
    • Thread Starter

    Ah , I see what you mean. Thanks for the advice!


    Thanks for checking!!
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