# Math Help - field of quotients

1. ## field of quotients

Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.

2. 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.
Well, there exists a homomorphism $\phi: R \mapsto R'$. 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 $R'$ is of the form $a\phi$ for some $a \in R$. Now, use the fact that this is a homomorphism, so $(a\phi)(b\phi) = \ldots$.