In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
In mathematics, and more specifically in abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms.
Mathematics is the study of topics such as quantity, structure, space, and change.
What is a Module? (Abstract Algebra) by Socratica
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 and a multiplication is defined between elements of the ring and elements of the module.
A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars.
Joomla Tutorial: Working With Modules by Incredible Tutorials
Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is compatible with the ring multiplication.
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
Modules are very closely related to the representation theory of groups.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.