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Math Help - field of quotients

  1. #1
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    field of quotients

    Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by bookie88 View Post
    Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.
    Well, there exists a homomorphism \phi: R \mapsto R'. It just so happens that this is an isomorphism, but that is not reeeeeeeeally needed. You need surjectivity, but not injectivity.

    So, as this is an isomorphism we have that every element of R' is of the form a\phi for some a \in R. Now, use the fact that this is a homomorphism, so (a\phi)(b\phi) = \ldots.
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