nLab quotient module

1. Idea

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.

2. Definition

Thoughout let $R$ be some ring. Write $R$Mod for the category of modules over $R$. Write $U:R Mod \to$ Set for the forgetful functor that sends a module to its underlying set.

Definition 2.1. 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 $N$.

3. Properties

Equivalent characterizations

Proposition 3.1. The quotient module is equivalently the cokernel of the inclusion in $R$Mod

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

Proposition 3.2. 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 $N$.

4. Related concepts

• quotient group

• quotient bimodule

• quotient ring

• quotient object