Definition:Field of Quotients/Definition 3

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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 $\varphi = \bar \varphi \circ \iota$
That is, the following diagram commutes:
$\begin{xy}\xymatrix@+1em@L+2px{D \ar[r]^\iota \ar[dr]_\varphi & F \ar[d]^{\exists_1 \bar \varphi} \\ & E}\end{xy}$


Also known as

Since the construction of the field of quotients $F$ from an integral domain $D$ mirrors the construction of the rationals from $\Z$, $F$ is sometimes called the field of fractions or fraction field of $D$.

Some sources prefer the term quotient field, but this can cause confusion with similarly named but unrelated concepts.

Common notations include $\map {\operatorname {Frac} } D$, $\map Q D$ and $\map {\operatorname {Quot} } D$.


Also see


Sources