Submodule

Definition

Let $R$ be a ring with identity element and let $M$ be a left module over $R$. A submodule of $M$ is a subset $N \subseteq M$ such that:

1. $N$ is a subgroup of the abelian group $(M, +)$.
2. $rx \in N$ for all $r \in R$ and $x \in N$.

Remarks

A submodule is a module over $R$ with the restricted operations. The submodules of $R$, as a left module over itself, are the left ideals of $R$. The $\Z$-submodules of an abelian group are its subgroups, and over a field the submodules are the subspaces. Intersections and sums of submodules are submodules.

Explanations

L1 human

A submodule is a module inside a bigger module. It contains the sums and the scalar multiples of its elements. For example, the even integers are a submodule of $\Z$ over $\Z$.