nLab
quotient module

Definition

Thoughout let $R$ be some ring. Write $R$Mod for the category of module over $R$. Write $UR\mathrm{Mod}\to$ Set for the forgetful functor that sends a module to its underlying set.

Definition

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

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

Related concepts

• quotient group