# Math Help - Rings, ID, and Fields

1. ## 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.

2. 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

3. Originally Posted by bookie88
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.