 Submodule

A submodule $N$ of a left module $M$ is a nonempty subset closed under addition, additive inverses and the scalar action of the ring $R$. Equivalently, it contains zero and is closed under expressions $rn+s{n}^{\prime }$ for $r,s\in R$ and $n,n^{\prime }\in N$.

For example, $2\mathbb{Z}$ is a submodule of the integer module $\mathbb{Z}$, and its quotient module is $\mathbb{Z}/2\mathbb{Z}$. The empty subset is excluded: closure conditions stated without nonemptiness would hold vacuously there, but it is not an additive group.