Module

A module is a vector-space-like structure whose scalars come from a ring rather than a field, the central object of module theory that unifies abelian groups, vector spaces, and representations.

Definition

A module over a ring R is an abelian group equipped with a scalar multiplication by elements of R that is distributive and associative and respects the identity, generalizing vector spaces to ring coefficients.

Scope

This topic covers the definition of a module over a ring, submodules and quotient modules, module homomorphisms, generators and relations, cyclic and finitely generated modules, and the isomorphism theorems, together with the basic examples of abelian groups and vector spaces as modules.

Core questions

• How does a module generalize a vector space and an abelian group?
• What are submodules, quotient modules, and module homomorphisms?
• How is a module presented by generators and relations?
• Why can modules fail to have a basis?

Key theories

Modules unify familiar structures

A module over a field is a vector space and a module over the integers is an abelian group, so module theory treats these and group-ring representations within a single framework.

Isomorphism theorems for modules

Module homomorphisms factor through quotients by their kernels, and the correspondence and isomorphism theorems carry over from groups and rings, organizing the structure of submodules and quotients.

Generators and relations

Every module is a quotient of a free module, so it is presented by generators and relations; the failure of relations to vanish is exactly what distinguishes general modules from vector spaces.

Clinical relevance

Modules are the common language for many algebraic structures: ideals and quotient rings, abelian groups, representations of groups and algebras, and the homology and cohomology groups of topology are all modules, so module theory provides tools that apply across mathematics.

History

The module concept generalized Dedekind's modules of algebraic numbers and the abelian groups of nineteenth-century arithmetic, and Emmy Noether placed it at the center of algebra in the 1920s by recognizing that ideals, quotients, and representations are all modules over suitable rings.

Key figures

• Emmy Noether
• Richard Dedekind
• Wolfgang Krull

Seminal works

• dummit2004
• lang2002
• atiyah1969

Frequently asked questions

Why is a module like a vector space with a ring of scalars?

The axioms are identical to those of a vector space except that scalars come from a ring rather than a field. Because ring elements need not be invertible, modules can have torsion and relations that no vector space exhibits.

What familiar objects are modules?

Abelian groups are modules over the integers, vector spaces are modules over fields, and ideals of a ring are modules over that ring. This is why a single theory of modules can address so many algebraic settings at once.