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Math Help - Rings, ID, and Fields

  1. #1
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    Rings, ID, and Fields

    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
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    Let  \phi : R \rightarrow R'
    If R is commumative then
    \forall a,b \in R: ab = ba
    \Rightarrow \phi(ab) = \phi(ba)
    Also \phi(ab) = \phi(a)\phi(b) and \phi(ba) = \phi(b)\phi(a)
    and \phi is bijective.

    This should be enough for you to see why that R' is communative
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  3. #3
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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.
    Let \phi:R\to R' denote an isomorphism, and x,y\in R'. Then there are a,b\in R with \phi(a)=x and \phi(b)=y. Furthermore, xy=\phi(a)\phi(b)=\phi(ab)=\phi(ba)=\phi(b)\phi(a)  =yx. So R' is commutative.

    EDIT: Looks like someone beat me to it. Oh well.
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