Kernel of a ring homomorphism

The kernel of a ring homomorphism $\varphi:R\to S$ is

i.e. the preimage of the additive identity of $S$.

The kernel is always an ideal of $R$ (indeed, a two-sided ideal), and it measures injectivity: $\varphi$ is a monomorphism iff $\ker(\varphi)=\{0\}$. Kernels are the basic input to forming quotient rings and proving isomorphism theorems.

Examples:

• For the reduction map $\mathbb Z\to \mathbb Z/n\mathbb Z$, the kernel is $n\mathbb Z$.
• For evaluation $\mathrm{ev}_c:k[x]\to k$, $\ker(\mathrm{ev}_c)=(x-c)$.
• The inclusion $\mathbb Z\hookrightarrow \mathbb Q$ has kernel $\{0\}$.