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.

Let $a + N\in M / N$ denote the coset of $N$ by $a\in M$.

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


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

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

Also see

 * Definition:Quotient Vector Space