nLab
quotient module

Contents

Definition

Thoughout let RR be some ring. Write RRMod for the category of module over RR. Write URModU R Mod \to Set for the forgetful functor that sends a module to its underlying set.

Definition

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

Properties

Equivalent characterizations

Proposition

The quotient module is equivalently the cokernel of the inclusion in RRMod

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

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

Created on September 11, 2012 10:10:33 by Urs Schreiber (82.169.65.155)