Ring monomorphism

Such a map identifies $R$ with the subring $\varphi(R)\subseteq S$. Equivalently, its kernel is zero.

• The inclusion $\mathbb Z\hookrightarrow\mathbb Q$ is a ring monomorphism.
• The inclusion $k[x]\hookrightarrow k[x,y]$ is a ring monomorphism.
• The reduction map $\mathbb Z\to\mathbb Z/n\mathbb Z$ is not a monomorphism for $n\ge2$.