Ring epimorphism - Knowlpedia

A ring epimorphism is a ring homomorphism $\varphi:R\to S$ such that for every ring $T$ and homomorphisms $g,h:S\to T$ ,

g\circ\varphi=h\circ\varphi\quad\Longrightarrow\quad g=h.

Thus $\varphi$ is an epimorphism in the category of rings.

Remarks

Every surjective ring homomorphism is an epimorphism, but the converse fails. In the category of commutative unital rings, for example, the localization map $\mathbb Z\to\mathbb Q$ is an epimorphism but is not surjective.

Examples

The quotient map $R\to R/I$ , $r\mapsto r+I$ , is a ring epimorphism.
The evaluation map $k[x]\to k$ , $f\mapsto f(c)$ , is a surjective ring epimorphism for every $c\in k$ .
The localization $\mathbb Z\to\mathbb Q$ is a nonsurjective ring epimorphism.