Module (mathematics)
Definition and Overview
A module in mathematics is a fundamental concept in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module.
Basic Concepts and Properties
A module consists of a set M equipped with two operations, analogous to the addition and scalar multiplication in vector spaces. The first operation is an abelian group operation, and the second operation is a scalar multiplication which interacts with the ring multiplication.
Abelian Group Structure
The set M is an abelian group under the first operation. This means that the operation is associative, there is an identity element, every element has an inverse, and the operation is commutative.
Scalar Multiplication
The scalar multiplication is a function from the ring R and the set M to M. It is required to satisfy the following conditions: 1. For all a, b in R and m in M, a*(b*m) = (a*b)*m. 2. For all a, b in R and m in M, (a + b)*m = a*m + b*m. 3. For all a in R and m, n in M, a*(m + n) = a*m + a*n. 4. For all m in M, 1*m = m.
Types of Modules
There are various types of modules defined over different types of rings. 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 generators. Every vector space is a free module, but not every free module is a vector space.
Projective Modules
A projective module is a direct summand of a free module. It has the property that every short exact sequence of modules that it is a part of splits.
Injective Modules
An injective module is a module that has the property that every short exact sequence of modules that it is a part of splits.
Flat Modules
A flat module is a module that preserves exact sequences under tensor product.
Module Homomorphisms and Isomorphisms
A module homomorphism is a function between two modules that preserves the module operations. If a module homomorphism is bijective, it is called a module isomorphism, and the two modules are said to be isomorphic.
Applications and Importance
Modules are a central object of study in abstract algebra and are used in various areas of mathematics. They are used in the study of representation theory, algebraic topology, and algebraic geometry, among others.