Kernel of a module homomorphism

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

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 $f:M\to 0$.