Ring monomorphism

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

Monomorphisms identify $R$ with a subring of $S$ up to isomorphism; equivalently, they are the homomorphisms with trivial kernel . In many contexts, one suppresses $\varphi$ and views $R$ as sitting inside $S$.

Examples:

• The inclusion $\mathbb Z\hookrightarrow \mathbb Q$ is a ring monomorphism.
• The map $k[x]\hookrightarrow k[x,y]$ sending $f(x)\mapsto f(x)$ is a ring monomorphism.
• The reduction map $\mathbb Z\to \mathbb Z/n\mathbb Z$ is not a monomorphism for $n\ge 2$.