Kernel of a module homomorphism

Let $f:M\to N$ be a module homomorphism. The kernel of $f$ is

\ker(f)=\{m\in M: f(m)=0\}.

It is a submodule, as recorded in kernels are submodules.

Kernels measure injectivity: $f$ is injective iff $\ker(f)=0$ . They also define the notion of exactness (see exact sequences, where kernels match images).

Examples

For $f:\mathbb Z^2\to\mathbb Z$ given by $f(a,b)=a+b$ , one has $\ker(f)=\{(t,-t):t\in\mathbb Z\}$ .
For $f:\mathbb Z\to\mathbb Z$ given by $f(n)=kn$ , the kernel is $0$ if $k\ne 0$ and all of $\mathbb Z$ if $k=0$ .
(Edge case) If $N=0$ , then $\ker(f)=M$ for every $f:M\to 0$ .