Submodule criterion

Submodule criterion: Let $M$ be an $R$-module and let $N\subseteq M$ be a subset. Then $N$ is a submodule of $M$ if and only if:

1. $0\in N$,
2. for all $x,y\in N$ we have $x-y\in N$, and
3. for all $r\in R$ and $x\in N$ we have $rx\in N$.

This is the standard test for submodules obtained by unpacking the module axioms for modules over a ring .