Isomorphism Field of Fractions

Question

Trying to figure out if you have integral domains A and B that are isomorphic then their respective field of fractions,F and K, are also isomorphic by $\theta(ab^{-1})=\theta(a)\theta(b)^{-1}$. How can you $\theta$ is an isomorphism? For one-to-one I figure show ker$\theta$=0 and use pigeonhole to show onto but not sure. And for addition would showing $\theta(ab^{-1} + cd^{-1}) = \theta(ad+bc)(bd)^{-1})$ be on the right track. Any help appreciated.

Answer 1

Use the universal property of fraction fields: $ $ if $\,D\,$ is a domain with fraction field $\,F\,$ then any ring injection $\,h:D\to K\,$ into a field $\,K\,$ extends uniquely to $\,\hat h : F\to K,\,$ via $\,\hat h (a/b) = h(a)h(b)^{-1}.$