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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- Feb 9th 2010, 05:44 AMbookie88field of quotients
Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.

- Feb 9th 2010, 08:17 AMSwlabr
Well, there exists a homomorphism $\displaystyle \phi: R \mapsto R'$. 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 $\displaystyle R'$ is of the form $\displaystyle a\phi$ for some $\displaystyle a \in R$. Now, use the fact that this is a homomorphism, so $\displaystyle (a\phi)(b\phi) = \ldots$.