For $N$ a module (over some ring$R$) and $S \hookrightarrow N$ a submodule, then the corresponding quotient module$N/S$ is the module where all elements in $N$ that differ by an element in $S$ are identified.

If the ring $R$ is a field then $R$-modules are called vector spaces and quotient modules are called quotient vector spaces.

For $i : K \hookrightarrow N$ a submodule, the quotient module$\frac{N}{K}$ is the quotient group of the underlying groups, equipped with the $R$-action induced by that on $N$.

Properties

Equivalent characterizations

Proposition

The quotient module is equivalently the cokernel of the inclusion in $R$Mod

$\frac{N}{K} \simeq coker(i)
\,.$

Proposition

The quotient module is equivalently the quotient object of the congruence$N \oplus K \to N \oplus N$ given by projection on the first factor and by addition in $N$.