nLab quotient module

Context

Algebra

higher algebra

universal algebra

Contents

Definition

Thoughout let $R$ be some ring. Write $R$Mod for the category of module over $R$. Write $U R Mod \to$ 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

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

Revised on December 22, 2015 15:02:32 by Urs Schreiber (195.37.209.180)