Module theory MOC

Module

A left-module 𝑉 over a ring 𝑅 is an abelian group 𝑉 together with an action of 𝑅 on 𝑉 that is distributive and 𝑅-linear. #m/def/module Unfurling this definition, being abelian group implies the following for 𝑣,𝑢,𝑤 ∈𝑉

1. (𝑣 +𝑢) +𝑤 =𝑣 +(𝑢 +𝑤)
2. 𝑣 +0 =𝑣
3. 𝑢 +𝑣 =𝑣 +𝑢

while 𝑅-linearity gives the following for 𝑣,𝑢 ∈𝑉 and 𝜆,𝜇 ∈𝑅

4. 1𝑣 =𝑣
5. (𝜇𝜆)𝑣 =𝜇(𝜆𝑣)
6. 𝜆(𝑢 +𝑣) =𝜆𝑢 +𝜆𝑣
7. (𝜇 +𝜆)𝑣 =𝜇𝑣 +𝜆𝑣

A right-module satisfies these properties mutatis mutandis with scalar multiplication written on the right.¹ Thus a module is a generalization of a vector space, which is just a module over a field. This small change has far-reaching implications, for example the existence of Torsion.

Further terminology

• Finitely generated module

Properties

• A module inherits linear structure from the underlying ring

Examples

• Vector space
• Let 𝐼 ⊴𝑅 be an ideal. Then 𝐼 is an 𝑅-submodule of 𝑅.
• Let 𝑇 :𝑅 be a ring extension. Then 𝑇 is an 𝑅-module.

#state/tidy | #lang/en | #SemBr