Quotient module

Let $M$ be a left $R$-module and let $N\le M$ be a submodule . Define an equivalence relation on $M$ by $m\sim m'$ iff $m-m'\in N$. The quotient module $M/N$ is the quotient set of equivalence classes, written $m+N$, with operations

These operations are well-defined precisely because $N$ is closed under subtraction and scalar multiplication.

The construction is characterized by the universal property of the quotient module : maps out of $M$ that kill $N$ factor uniquely through $M/N$.

Examples:

• For $M=\mathbb Z^2$ and $N=2\mathbb Z^2$, the quotient $M/N$ has four elements and is isomorphic (as a $\mathbb Z$-module) to $(\mathbb Z/2\mathbb Z)^2$.
• For a ring $R$ and ideal $I\lhd R$, the quotient $R/I$ is a quotient module of the left $R$-module $R$.
• (Edge case) If $N=M$, then $M/N$ is the zero module; if $N=0$, then $M/N\cong M$.