Module (mathematics)

A module is a mathematical structure in which elements can be added and multiplied by scalars belonging to a ring. It generalizes a vector space by replacing its scalar field with a ring, whose nonzero elements need not have multiplicative inverses. Modules also generalize abelian groups, which are precisely modules over the ring of integers. This common framework connects linear algebra with the study of groups, rings, and their representations. (math.buffalo.edu)

Definition and conventions

Let $R$ be an associative ring with identity $1_R$. A left $R$-module consists of an abelian group $(M,+)$ and a scalar multiplication

satisfying, for all $r,s\in R$ and $m,n\in M$,

The first two axioms express distributivity; the third makes scalar multiplication compatible with ring multiplication; the fourth says that the identity scalar acts as the identity. They imply

Here modules are assumed to satisfy the identity axiom; some treatments allow nonunital modules or rings without identity. (people.math.osu.edu)

A right $R$-module instead has multiplication $mr$, with

Left and right modules must be distinguished when $R$ is noncommutative. A right $R$-module can be viewed as a left module over the opposite ring, whose multiplication reverses that of $R$. For a commutative ring, left and right modules correspond by setting $mr=rm$. (math.buffalo.edu)

Fundamental examples

Vector spaces. A module over a field is exactly a vector space over that field. Thus familiar spaces such as $\mathbb R^n$ and $\mathbb C^n$ are modules. The essential difference is that a general ring does not permit division by every nonzero scalar. (math.buffalo.edu)

Abelian groups. Every abelian group has a unique $\mathbb Z$-module structure: for a positive integer $n$, the expression $nm$ means adding $m$ to itself $n$ times, and negative integers act using additive inverses. Conversely, the additive group of any $\mathbb Z$-module is abelian. Group homomorphisms between abelian groups automatically preserve this scalar multiplication. (jmilne.org)

Rings and ideals. A ring is a left module over itself by multiplication, and its left ideals are submodules. For a commutative ring $R$, any ideal $I$ is an $R$-module, as is the quotient $R/I$, with

The coordinate module $R^n$ has componentwise addition and scalar multiplication. (faculty.niu.edu)

Linear operators. If $V$ is a vector space over a field $k$ and $T:V\to V$ is a linear operator, then $V$ becomes a module over the polynomial ring $k[x]$ by defining

In particular, $xv=T(v)$. Polynomial identities satisfied by $T$ become relations in this module. This construction allows questions about linear operators to be studied through module structure. (people.math.osu.edu)

Submodules, quotients, and homomorphisms

A submodule $N\subseteq M$ is an additive subgroup closed under scalar multiplication. Given a submodule, the quotient module $M/N$ consists of additive cosets, with the well-defined action

These constructions extend linear subspaces and quotient vector spaces to arbitrary scalar rings. (stacks.math.columbia.edu)

An $R$-module homomorphism $f:M\to N$ preserves both operations:

Its kernel and image are submodules, and the first isomorphism theorem gives

A bijective module homomorphism is a module isomorphism. Modules over a fixed ring and their homomorphisms form a category, usually denoted $R\text{-}\mathrm{Mod}$ for left modules. (stacks.math.columbia.edu)

An exact sequence records how homomorphisms fit together: at each intermediate term, the image of the incoming map equals the kernel of the outgoing map. A short exact sequence

identifies $A$ with a submodule of $B$ and $C$ with the resulting quotient. Unlike short exact sequences of vector spaces, such sequences need not split into a direct-sum decomposition. (jmilne.org)

Generators, bases, and presentations

A subset $S\subseteq M$ generates $M$ if every element is a finite linear combination of elements of $S$, with coefficients in $R$. A finitely generated module has a finite generating set. Equivalently, it is the image of a surjection $R^n\to M$ for some finite $n$. A module generated by one element is called cyclic. In this terminology, “finite module” often means finitely generated, not finite as a set. (stacks.math.columbia.edu)

A free module has a basis: a generating set for which every element has a unique finite expression as a linear combination. Equivalently, it is isomorphic to

for some index set $I$. The parentheses indicate that only finitely many coordinates may be nonzero. This direct sum differs from the unrestricted direct product when $I$ is infinite. (stacks.math.columbia.edu)

Every module is a quotient of a free module, but not every module is free. For example, $\mathbb Z/n\mathbb Z$, with $n>1$, is cyclic as a $\mathbb Z$-module but has no basis: every element is annihilated by the nonzero scalar $n$, preventing any nonempty set from being linearly independent. (faculty.niu.edu)

A module is finitely presented if it has an exact sequence

with finite $m,n$. Informally, it has finitely many generators and finitely many generating relations. For a commutative ring, $A$ can be represented by a matrix over $R$, and

Finite presentation implies finite generation, but the converse fails over general rings. (stacks.math.columbia.edu)

Torsion and classification over principal ideal domains

Suppose $R$ is an integral domain. An element $m\in M$ is a torsion element if $rm=0$ for some nonzero $r\in R$. These elements form the torsion submodule $T(M)$. A module is torsion-free if $T(M)=0$, and is a torsion module if $T(M)=M$. For $\mathbb Z$-modules, torsion elements are exactly the elements of finite additive order. (dlk53.github.io)

A particularly strong classification holds over a principal ideal domain (PID), an integral domain in which every ideal is generated by one element. Every finitely generated module over a PID has a decomposition

where the $d_i$ are nonzero nonunits and

The integer $r$ and the ideals $(d_i)$ are uniquely determined. The first summand is free; the remaining summands comprise the torsion part. In particular, every finitely generated torsion-free module over a PID is free. (dlk53.github.io)

For $R=\mathbb Z$, this yields the classification of finitely generated abelian groups. For $R=k[x]$, applied to the module associated with a linear operator, it yields rational canonical form and, when the relevant polynomials split over $k$, Jordan canonical form. These are two instances of the same module-theoretic structure theorem. (math.buffalo.edu)

Important classes of modules

A projective module $P$ has a lifting property: whenever $q:N\to L$ is surjective and $f:P\to L$ is a homomorphism, there exists $g:P\to N$ with $qg=f$. Equivalently, $P$ is a direct summand of a free module. Every free module is projective, but projective modules need not be free. (stacks.math.columbia.edu)

A flat module is one for which tensoring preserves exact sequences. Free modules are flat, and so are projective modules. Over a commutative ring, a finitely presented flat module is projective; mere finite generation is not sufficient in general. (jmilne.org)

A module is simple if it is nonzero and has no nonzero proper submodules. A semisimple module is a direct sum of simple modules. These notions are central in representation theory, where simple modules describe irreducible representations. (jmilne.org)

A Noetherian module satisfies the ascending chain condition on submodules: every increasing chain eventually stabilizes. Equivalently, every submodule is finitely generated. Over a Noetherian commutative ring, every finitely generated module is Noetherian and finitely presented. These conditions control the size of systems of generators and relations. (jmilne.org)

Tensor products and geometric applications

For modules $M,N$ over a commutative ring, the tensor product $M\otimes_RN$ is characterized by a universal property for $R$-bilinear maps. Its defining relations include

Tensor products are right exact, but they do not generally preserve injections; flatness specifies when this additional preservation holds. For noncommutative rings, tensoring a right module with a left module produces an abelian group, with further module structures requiring additional compatible actions. (stacks.math.columbia.edu)

A homomorphism of commutative rings $R\to S$ allows scalars to be extended:

Localization is an important instance. For a multiplicatively closed subset $U\subseteq R$,

It makes the elements of $U$ invertible and permits module properties to be examined locally. (stacks.math.columbia.edu)

In algebraic geometry, finitely generated projective modules over a commutative ring correspond to finite locally free sheaves on its affine spectrum. Their local freeness means that, after suitable localizations, they become finite free modules. Their ranks may vary between connected components, unlike the single dimension associated with a vector space over a field. (stacks.math.columbia.edu)

In group representation theory, a linear representation of a group $G$ over a field $k$ is equivalently a module over the group algebra $k[G]$: group elements act as linear operators, and their action extends linearly to the algebra. Thus module homomorphisms encode maps compatible with the group action, while submodules encode invariant subspaces. (jmilne.org)

References

1. Notes on Algebramath.buffalo.edu
2. Lecture 29: Modulespeople.math.osu.edu
3. Modulesfaculty.niu.edu
4. Group Theory — J. S. Milnejmilne.org
5. Basic notions — The Stacks Projectstacks.math.columbia.edu
6. Finite modules and finitely presented modules — The Stacks Projectstacks.math.columbia.edu
7. Modules over PIDsdlk53.github.io
8. Projective modules — The Stacks Projectstacks.math.columbia.edu
9. Finite projective modules — The Stacks Projectstacks.math.columbia.edu
10. Tensor products — The Stacks Projectstacks.math.columbia.edu
11. Modules of finite presentation — The Stacks Projectstacks.math.columbia.edu
12. A Primer of Commutative Algebra — J. S. Milnejmilne.org