Module theory MOC

Module

A module (𝑀,𝑅, +, ⋅) over a ring 𝑅 is an abelian group (𝑀, +) together with an action ( ⋅) of 𝑅 on 𝑉 that is distributive and 𝑅-linear. module Explicitly, a left-module 𝑀 over 𝑅 satisfies the following for any 𝑥,𝑦,𝑧 ∈𝑀 and 𝜇,𝜆 ∈𝑅

1. (𝑣 +𝑢) +𝑤 =𝑣 +(𝑢 +𝑤)
2. 𝑣 +0 =𝑣
3. 𝑢 +𝑣 =𝑣 +𝑢
4. 1𝑣 =𝑣
5. (𝜇𝜆)𝑣 =𝜇(𝜆𝑣)
6. 𝜆(𝑢 +𝑣) =𝜆𝑢 +𝜆𝑣
7. (𝜇 +𝜆)𝑣 =𝜇𝑣 +𝜆𝑣

whereas a right-module satisfies the same properties 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.

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tidy | en | SemBr

Footnotes

1. If 𝑅 is a commutative ring the concepts of left- and right-modules coïncide, but otherwise there is a distinction between left- and right-scalar multiplication. ↩