Definition:Quotient Module

Definition
Let $M$ be an $R$-module and $M^\ast$ be the underlying group.

Let $N$ be a submodule of $M$ and $N^\ast$ be the underlying subgroup.

Let $a + N$ denote the coset of the quotient group $M^\ast / N^\ast$.

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


 * $\forall r \in R, \forall a \in M / N: r \left({a + N^\ast}\right) := r a + N^\ast$

Then $(M^\ast / N^\ast,\circ)$ is a quotient $R$-module.