# Thread: 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 $\displaystyle \phi : R \rightarrow R'$
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

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 $\displaystyle \phi:R\to R'$ 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.