Definition:Field of Quotients/Definition 3

Definition
Let $D$ be an integral domain.

A field of quotients of $D$ is a pair $\struct {F, \iota}$ where:
 * $(1): \quad$ $F$ is a field
 * $(2): \quad$ $\iota : D \to F$ is a ring monomorphism
 * $(3): \quad$ it satisfies the following universal property:
 * For every field $E$ and for every ring monomorphism $\varphi: D \to E$, there exists a unique field homomorphism $\bar \varphi: F \to E$ such that $\varphi = \bar \varphi \circ \iota$, i.e. the following diagram commutes:
 * $\begin{xy}\xymatrix@+1em@L+2px{D \ar[r]^\iota \ar[dr]_\varphi & F \ar[d]^{\exists ! \bar \varphi} \\ & E}\end{xy}$

Also see

 * Equivalence of Definitions of Field of Quotients