Definition:Quotient Module
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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 $N$.