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Submodule

Definition

If an abelian subgroup $N$ of a module $M$ over a ring $R$ is such that $r.N \subseteq N$ for all $r \in R$, then $N$ is a submodule of $M$.

Criterion

The following together are necessary and sufficient:

• $N \subseteq M$ is a subgroup - see the subgroup criterion
• $\forall r \in R; n \in N: r.n \in N$

Properties

Given any submodule $N \le M$ we can define a quotient module $M/N$.

Examples

The torsion elements $M^{tor}$ of a module (over an integral domain) form a submodule. The quotient module $M/M^{tor}$ is torsion-free.

all these pages adapted with probably insufficient credit from my university's lecture notes