nLab
quotient module

Contents

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 modules over RR. Write U:RModU: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.

Last revised on December 11, 2020 at 17:52:43. See the history of this page for a list of all contributions to it.