Definition:Quotient Module

Definition

Let $M$ be an $R$-module and $\struct {M, +}$ be the underlying abelian group.

Let $N$ be a submodule of $M$ and $\struct {N, +}$ be the underlying abelian group.

Let $M / N$ be the quotient group.

Define the $R$-action on $M / N$ as:

$\forall r \in R, \forall a \in M : r \circ \paren {a + N} := r a + N$

where $a + N$ denotes the coset of $N$ by $a$ in $M / N$.

Then $\struct {M / N, +, \circ}$ is the quotient $R$-module of $M$ by $N$.