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**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

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**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.
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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

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**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.
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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.

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**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.

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**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.