Module Theory
Introduction
Module theory is a branch of mathematics that studies modules over a ring. A module is a mathematical structure that extends the notion of vector space and includes it as a special case. The study of modules and their properties forms a key part of abstract algebra.
Definition
A module over a ring is a generalization of the notion of vector space over a field, where the corresponding scalars in the module are the elements of an arbitrary given ring (and not necessarily a field), and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module. Thus, the module structure is a natural generalization of the concept of linear algebra.
Types of Modules
There are various types of modules defined over a ring. Some of the most common types include:
Free Modules
A free module is a module that has a basis, or a set of linearly independent vectors that span the module. This is analogous to a free vector space in linear algebra.
Projective Modules
A projective module is a direct summand of a free module. It has the property that any short exact sequence of modules that it appears in splits.
Injective Modules
An injective module is a module that has the property that any exact sequence of modules that it appears in can be extended to a longer exact sequence.
Flat Modules
A flat module is a module that preserves exact sequences under the operation of tensor product.
Properties of Modules
Modules over a ring have many properties that are analogous to properties of vector spaces. Some of these properties include:
Submodules
A submodule of a module is a subset of the module that is closed under the operations of the module. This is analogous to a subspace in a vector space.
Quotient Modules
A quotient module is a module formed by "dividing" a module by a submodule. This is analogous to a quotient space in a vector space.
Homomorphisms
A homomorphism between two modules is a function that preserves the module operations. This is analogous to a linear map in a vector space.
Applications of Module Theory
Module theory has applications in various areas of mathematics, including:
- Algebraic topology, where chain complexes of modules are used to define homology and cohomology groups. - Algebraic geometry, where modules of sections of line bundles are used to study algebraic varieties. - Representation theory, where modules over a group ring are used to study representations of groups.
See Also
- Ring theory - Category theory - Homological algebra