Submodule

Let $M$ be a left $R$-module . A submodule of $M$ is a subset $N\subseteq M$ such that:

1. $0\in N$,
2. $n_1,n_2\in N \Rightarrow n_1-n_2\in N$,
3. $r\in R$ and $n\in N \Rightarrow rn\in N$.

Equivalently, $N$ is an additive subgroup of $(M,+)$ that is stable under scalar multiplication. Practical closure tests are summarized in the submodule criterion .

Examples:

• In the $\mathbb Z$-module $M=\mathbb Z^2$, the set $N=\{(2a,2b):a,b\in\mathbb Z\}$ is a submodule.
• If $R$ is a ring, any ideal $I\lhd R$ is a submodule of the left $R$-module $R$.
• (Edge case) The sets $\{0\}$ and $M$ are always submodules; a module is simple exactly when these are the only submodules.