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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- Mar 24th 2010, 09:13 AMbookie88Rings, ID, and Fields
Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then r' is commutative.

- Mar 24th 2010, 09:24 AMHaven
Let $\displaystyle \phi : R \rightarrow R'$

If R is commumative then

$\displaystyle \forall a,b \in R: ab = ba$

$\displaystyle \Rightarrow \phi(ab) = \phi(ba)$

Also $\displaystyle \phi(ab) = \phi(a)\phi(b)$ and $\displaystyle \phi(ba) = \phi(b)\phi(a)$

and $\displaystyle \phi$ is bijective.

This should be enough for you to see why that R' is communative - Mar 24th 2010, 09:31 AMhatsoff
Let $\displaystyle \phi:R\to R'$ denote an isomorphism, and $\displaystyle x,y\in R'$. Then there are $\displaystyle a,b\in R$ with $\displaystyle \phi(a)=x$ and $\displaystyle \phi(b)=y$. Furthermore, $\displaystyle xy=\phi(a)\phi(b)=\phi(ab)=\phi(ba)=\phi(b)\phi(a) =yx$. So $\displaystyle R'$ is commutative.

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