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
Objects
Types of 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
Morphisms
The morphisms of interest are module homomorphisms which are defined precisely the same as linear maps of vector spaces. We also have
-multilinear map for a commutative ring . -balanced product
External constructions
- Submodule, Quotient module
- Direct sum of modules
- Free module
- Tensor product of modules over a noncommutative ring