Given a ring_homomorphism $f:R\to S$ between rings $R$ and $S$, we say the kernel of $f$ is the collection of elements of $R$ which $f$ sends to the zero element of $S$.

Formally, it is $$\{r\in R\mid f(r)=0_S\}$$ where $0_S$ is the zero element of $S$.

Examples

• Given the "identity" (or "do nothing") ring homomorphism $id:Z\to Z$, which sends $n$ to $n$, the kernel is just $\left\{0\right\}$.
• Given the ring homomorphism $Z\to Z$ taking $n\mapsto n\left(\mathrm{mod}2\right)$ (using the usual shorthand for modular arithmetic), the kernel is the set of even numbers.

Properties

Kernels of ring homomorphisms are very important because they are precisely ideals. (Proof.) In a way, "ideal" is to "ring" as "subgroup" is to "group", and certainly subrings are much less interesting than ideals; a lot of ring theory is about the study of ideals.

The kernel of a ring homomorphism always contains $0$, because a ring homomorphism always sends $0$ to $0$. This is because it may be viewed as a group homomorphism acting on the underlying additive group of the ring in question, and the image of the identity is the identity in a group.

If the kernel of a ring homomorphism contains $1$, then the ring homomorphism sends everything to $0$. Indeed, if $f\left(1\right)=0$, then $f\left(r\right)=f(r\times 1)=f\left(r\right)\times f\left(1\right)=f\left(r\right)\times 0=0$.