Ring epimorphism

A ring epimorphism is a ring homomorphism $\varphi:R\to S$ that is surjective as a function.

Epimorphisms present $S$ as a quotient of $R$; the basic example is the natural projection onto a quotient ring . They are the appropriate maps for “presenting” rings by generators and relations.

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 surjective for any field $k$ and $c\in k$.
• The inclusion $\mathbb Z\hookrightarrow \mathbb Q$ is not an epimorphism (not surjective).