nLab
quotient module

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

For NN a module (over some ring RR) and SNS \hookrightarrow N a submodule, then the corresponding quotient module N/SN/S is the mdoule where all elements in NN that differ by n element in SS are identified.

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

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.

Revised on May 29, 2017 10:38:11 by Urs Schreiber (94.220.14.70)