Kernels are submodules

Let $f:M\to N$ be a homomorphism of $R$-modules. Its kernel

is an $R$-submodule of $M$.

Indeed, if $x,y\in\ker(f)$ and $r\in R$, then $f(x+y)=0$ and $f(rx)=rf(x)=0$. Thus the kernel is the canonical submodule used to form the quotient by the kernel.