[[Ring theory MOC]]
# Module theory MOC

**Module theory** isn't a standard region of mathematics, since the theory of modules is typically viewed as a small part of [[Ring theory MOC]] and [[Representation theory MOC]].
The theory of modules typically follows the process of generalising results from [[Linear algebra MOC]] from modules over fields to modules over more general rings as best as possible.

## Fundamentals

The object of interest is the [[Module]], which is an algebraic structure defined using a [[Ring]].
Unless we are dealing with a [[Module over a commutative ring]], it is necessary to distinguish between left-modules and right-modules.
The left modules over a given ring $R$ form a category $\lMod{R}$,
the right modules form a category $\rMod R$.
See [[Category of left modules]].

## Objects
### Types of module

- [[Reducibility of modules]]
- [[Noetherian module]]
- [[Finitely generated module]]
- [[Free module]]
- [[Projective module]]

#### By ring

- [[Module over a commutative ring]]
- [[Module over a unital associative algebra]]
- [[Vector space]] (module over a field)


### Properties of modules

- [[Rank of a module]] over an [[integral domain]]

### Additional structure

- [[Graded module]]

## Morphisms

The morphisms of interest are [[Module homomorphism|module homomorphisms]] which are defined precisely the same as linear maps of vector spaces. We also have

- $R$-[[multilinear map]] for a [[commutative ring]] $R$.
- $R$-[[balanced product]]

## External constructions

- [[Submodule]], [[Quotient module]]
- [[Direct sum of modules]]
- [[Free module]]
- [[Tensor product of modules over a noncommutative ring]]

### Commutative rings

- [[Tensor product of modules over a commutative ring]]

### Juggling multiple rings

- [[Induced module]]

## Bibliography

- https://agag-lassueur.math.rptu.de/~lassueur/en/teaching/COHOMSS18/CGSS18/Kap2.pdf

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