[[Module theory MOC]]
# Submodule

Let $(M, R, +, \cdot)$ be a (left) [[module]].
A **submodule** $N \leq M$ is a module under the same operations, #m/def/module 
i.e. $(N, +)$ is a [[subgroup]] such that $r \cdot n \in N$ for any $n \in N$ and $r \in R$.
Thus a submodule is an **invariant subspace** under the carried representation of $R$ (see [[invariant subspace]]).

## Examples

- Let $I \trianglelefteq R$ be an [[ideal]]. Then $I$ is an $R$-submodule of $R$.

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#state/tidy | #lang/en | #SemBr