# A doubt about the isomorphism between an integral domain and its field of quotients.

I am currently reading the proof of the fact that An integral domain can be imbedded into a field.

Let $a,b,c,d\in D$, where $D$ is an integral domain.

1. $(a,b)=(c,d)$ iff $ad=bc$
2. $(a,b)+(c,d)=(ad+bc,bd)$

$(a,b)$ here stands for $a/b$

The book (Topics in Algebra by Herstein) states that the mapping $\phi (a)=[a,1]$ is an isomorphism from $D$, which is the integral domain, to the field $F$, which is the field of quotients of $D$.

I don't understand how. For example, the field of quotients of $\Bbb{Z}$ would be $\Bbb{Q}$. Clearly one equivalence class in $\Bbb{Q}$ is $[1,2]$. What element of $\Bbb{Z}$ is mapped to $[1,2]$ through this mapping?

I'm not sure what Herstein's definitions are, but yeah, you're right that the mapping is not an isomorphism of $D$ with $F$ since it is not onto $F$. It is just an injective ring homomorphism of $D$ into $F$. (A real, surjective isomorphism would be nonsensical anyway since rings isomorphic to fields are fields, and your domain will not always be a field.)