# Fields as a reflective subcategory of integral domains?

A subcategory $\mathbf A$ is reflective subcategory of $\mathbf B$ if for every $B\in\mathbf B$ there exists an $A_B\in\mathbf A$ and a $\mathbf B$-morphism $r_B \colon B \to A_B$ such that: for any $A\in\mathbf A$ and any $\mathbf B$-morphism $f \colon B \to A$ there exists unique $\mathbf A$-morphism $\overline f \colon A_B \to A$ such that $\overline f\circ r_B=f$. This is equivalent to saying that the inclusion functor $E \colon \mathbf A \to \mathbf B$ has a left adjoint.

Current revision of the Wikipedia article on reflective subcategories claims that

The category of fields is a reflective subcategory of the category of integral domains. The reflector is the functor which sends each integral domain to its field of fractions.

I don't think this is true - an easy counterexample is the homomorphism $f \colon \mathbb Z \to\mathbb Z_2$ given by $f(n)=n\mod 2$. This homorphism obviously cannot be extended to $\overline f \colon \mathbb Q \to \mathbb Z_2$.

The above claim would be true if we considered the categories of integral domains only with injective ring homomorhpisms.

Am I correct? Did I miss something there?

The same problem has already been mentioned on the talk page of this Wikipedia article, so this is more-or-less just a sanity check.

• The article field of fractions states explicitly what the category of integral domains has to be here. Jun 6 '12 at 10:16
• Small typo in definition of $r_B$: domain should be $B$, not $A$. Jan 24 '16 at 12:31
• Thanks @JoshuaMeyers. Corrected now. Jan 24 '16 at 14:50