Wikipedia defines the field of fractions of a domain as
The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded.
What does "smallest" mean mathematically in this context? Is it possible to embed an integral domain in two different fields which have no elements in common?
Best Answer
"Smallest" means essentially that it satisfies the following universal property: if $R$ is a domain, and $K$ is any field for which there exists an injective ring homomorphism $f:R\to K$, then there is a unique ring homomorphism $g:\text{Frac}(R)\to K$ such that $f=g\circ i$, where $i$ is the natural inclusion of $R$ into its fraction field $\mathrm{Frac}(R)$.
As usual for a universal property, the object which satisfies it is unique up to unique isomorphism. Thus, if $L$ is a field in which $R$ embeds, and $L$ has the property that $L$ embeds in any other field in which $R$ embeds (this is the sense of "smallest"), then $L$ will necessarily be isomorphic to $\mathrm{Frac}(R)$.