Definition:Quotient Module

Definition
Let $M$ be a given $R$-module and $N$ a submodule of $M$. Let $a+N$ denote the coset of the quotient group $M/N$.

Define an operation $+$ the same way it is for the quotient group $M/N$

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

$\forall r\in R,\forall a\in M/N: r(a+N) := ra + n$

Then $M/N$ is a quotient $R$-module