# Module (mathematics)

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Generalization of vector spaces from fields to rings

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Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring Z {\displaystyle \mathbb {Z} } • Terminal ring 0 = Z / 1 Z {\displaystyle 0=\mathbb {Z} /1\mathbb {Z} } Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Integers modulo n • Ring of integers • p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} • p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} • Prüfer p-ring Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra v t e

Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Rng Semiring Near-ring Commutative ring Domain Integral domain Field Division ring Lie ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory Module-like Module Group with operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra v t e

In [mathematics](/source/Mathematics), a **module** is a generalization of the notion of [vector space](/source/Vector_space) in which the [field](/source/Field_(mathematics)) of [scalars](/source/Scalar_(mathematics)) is replaced by a (not necessarily [commutative](/source/Commutative_ring)) [ring](/source/Ring_(mathematics)). The concept of a *module* also generalizes the notion of an [abelian group](/source/Abelian_group), since the abelian groups are exactly the modules over the ring of [integers](/source/Integer).[1]

Like a vector space, a module is an additive abelian group, and scalar multiplication is [distributive](/source/Distributive_property) over the operations of addition between elements of the ring or module and is [compatible](/source/Semigroup_action) with the ring multiplication.

Modules are very closely related to the [representation theory](/source/Representation_theory) of [groups](/source/Group_(mathematics)). They are also one of the central notions of [commutative algebra](/source/Commutative_algebra) and [homological algebra](/source/Homological_algebra), and are used widely in [algebraic geometry](/source/Algebraic_geometry) and [algebraic topology](/source/Algebraic_topology).

## Introduction and definition

### Motivation

In a vector space, the set of [scalars](/source/Scalar_(mathematics)) is a [field](/source/Field_(mathematics)) and acts on the vectors by scalar multiplication, subject to certain axioms such as the [distributive law](/source/Distributive_law). In a module, the scalars need only be a [ring](/source/Ring_(mathematics)), so the module concept represents a significant generalization. In commutative algebra, both [ideals](/source/Ideal_(ring_theory)) and [quotient rings](/source/Quotient_ring) are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.

Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "[well-behaved](/source/Well-behaved)" ring, such as a [principal ideal domain](/source/Principal_ideal_domain). However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a [basis](/source/Basis_(linear_algebra)), and, even for those that do ([free modules](/source/Free_module)), the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique [rank](/source/Free_module#Definition)) if the underlying ring does not satisfy the [invariant basis number](/source/Invariant_basis_number) condition, unlike vector spaces, which always have a (possibly infinite) basis whose [cardinality](/source/Cardinality) is then unique. (These last two assertions require the [axiom of choice](/source/Axiom_of_choice) in general, but not in the case of [finite-dimensional](/source/Finite-dimensional) vector spaces, or certain well-behaved infinite-dimensional vector spaces such as [L*p* spaces](/source/Lp_space).)

### Formal definition

Suppose that *R* is a [ring](/source/Ring_(mathematics)), and 1 is its multiplicative identity. A **left *R*-module** *M* consists of an [abelian group](/source/Abelian_group) (*M*, +) and an operation **·** : *R* × *M* → *M* such that for all *r*, *s* in *R* and *x*, *y* in *M*, we have

- r ⋅ ( x + y ) = r ⋅ x + r ⋅ y {\displaystyle r\cdot (x+y)=r\cdot x+r\cdot y} ,

- ( r + s ) ⋅ x = r ⋅ x + s ⋅ x {\displaystyle (r+s)\cdot x=r\cdot x+s\cdot x} ,

- ( r s ) ⋅ x = r ⋅ ( s ⋅ x ) {\displaystyle (rs)\cdot x=r\cdot (s\cdot x)} ,

- 1 ⋅ x = x . {\displaystyle 1\cdot x=x.}

The operation · is called *scalar multiplication*. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in *R*. One may write *R**M* to emphasize that *M* is a left *R*-module. A **right *R*-module** *M**R* is defined similarly in terms of an operation · : *M* × *R* → *M*.

The qualificative of left- or right-module does not depend on whether the scalars are written on the left or on the right, but on the property 3: if, in the above definition, the property 3 is replaced by

- ( r s ) ⋅ x = s ⋅ ( r ⋅ x ) , {\displaystyle (rs)\cdot x=s\cdot (r\cdot x),}

one gets a right-module, even if the scalars are written on the left. However, writing the scalars on the left for left-modules and on the right for right modules makes the manipulation of property 3 much easier.

Authors who do not require rings to be [unital](/source/Unital_algebra) omit condition 4 in the definition above; they would call the structures defined above "unital left *R*-modules". In this article, consistent with the [glossary of ring theory](/source/Glossary_of_ring_theory), all rings and modules are assumed to be unital.[2]

An (*R*,*S*)-[bimodule](/source/Bimodule) is an abelian group together with both a left scalar multiplication · by elements of *R* and a right scalar multiplication ∗ by elements of *S*, making it simultaneously a left *R*-module and a right *S*-module, satisfying the additional condition (*r* · *x*) ∗ *s* = *r* ⋅ (*x* ∗ *s*) for all *r* in *R*, *x* in *M*, and *s* in *S*.

If *R* is [commutative](/source/Commutative_ring), then left *R*-modules are the same as right *R*-modules and are simply called *R*-modules. Most often the scalars are written on the left in this case.

## Examples

- If *K* is a [field](/source/Field_(mathematics)), then *K*-modules are called *K*-[vector spaces](/source/Vector_space) (vector spaces over *K*).

- If *K* is a field, and *K*[*x*] a univariate [polynomial ring](/source/Polynomial_ring), then a [*K*\[*x*\]-module](/source/Polynomial_ring#Modules) *M* is a *K*-module with an additional action of *x* on *M* by a group homomorphism that commutes with the action of *K* on *M*. In other words, a *K*[*x*]-module is a *K*-vector space *M* combined with a [linear map](/source/Linear_map) from *M* to *M*. Applying the [structure theorem for finitely generated modules over a principal ideal domain](/source/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain) to this example shows the existence of the [rational](/source/Rational_canonical_form) and [Jordan canonical](/source/Jordan_normal_form) forms.

- The concept of a **Z**-module agrees with the notion of an abelian group. That is, every [abelian group](/source/Abelian_group) is a module over the ring of [integers](/source/Integer) **Z** in a unique way. For *n* > 0, let *n* ⋅ *x* = *x* + *x* + ... + *x* (*n* summands), 0 ⋅ *x* = 0, and (−*n*) ⋅ *x* = −(*n* ⋅ *x*). Such a module need not have a [basis](/source/Basis_(linear_algebra))—groups containing [torsion elements](/source/Torsion_element) do not. (For example, in the group of integers [modulo](/source/Modular_arithmetic) 3, one cannot find even one element that satisfies the definition of a [linearly independent](/source/Linearly_independent) set, since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a [finite field](/source/Finite_field) is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)

- The [decimal fractions](/source/Decimal_fractions) (including negative ones) form a module over the integers. Only [singletons](/source/Singleton_(mathematics)) are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no [rank](/source/Rank_of_a_free_module), in the usual sense of linear algebra. However this module has a [torsion-free rank](/source/Torsion-free_rank) equal to 1.

- If *R* is any ring and *n* a [natural number](/source/Natural_number), then the [cartesian product](/source/Cartesian_product) *R**n* is both a left and right *R*-module over *R* if we use the component-wise operations. Hence when *n* = 1, *R* is an *R*-module, where the scalar multiplication is just ring multiplication. The case *n* = 0 yields the trivial *R*-module {0} consisting only of its identity element. Modules of this type are called [free](/source/Free_module) and if *R* has [invariant basis number](/source/Invariant_basis_number) (e.g. any commutative ring or field) the number *n* is then the rank of the free module.

- If M*n*(*R*) is the ring of *n* × *n* [matrices](/source/Matrix_(mathematics)) over a ring *R*, *M* is an M*n*(*R*)-module, and *e**i* is the *n* × *n* matrix with 1 in the (*i*, *i*)-entry (and zeros elsewhere), then *e**i**M* is an *R*-module, since *re**i**m* = *e**i**rm* ∈ *e**i**M*. So *M* breaks up as the [direct sum](/source/Direct_sum) of *R*-modules, *M* = *e*1*M* ⊕ ... ⊕ *e**n**M*. Conversely, given an *R*-module *M*0, then *M*0⊕*n* is an M*n*(*R*)-module. In fact, the [category of *R*-modules](/source/Category_of_modules) and the [category](/source/Category_(mathematics)) of M*n*(*R*)-modules are [equivalent](/source/Equivalence_of_categories). The special case is that the module *M* is just *R* as a module over itself, then *R**n* is an M*n*(*R*)-module.

- If *S* is a [nonempty](/source/Empty_set) [set](/source/Set_(mathematics)), *M* is a left *R*-module, and *M**S* is the collection of all [functions](/source/Function_(mathematics)) *f* : *S* → *M*, then with addition and scalar multiplication in *M**S* defined pointwise by (*f* + *g*)(*s*) = *f*(*s*) + *g*(*s*) and (*rf*)(*s*) = *rf*(*s*), *M**S* is a left *R*-module. The right *R*-module case is analogous. In particular, if *R* is commutative then the collection of *R-module homomorphisms* *h* : *M* → *N* (see below) is an *R*-module (and in fact a *submodule* of *N**M*).

- If *X* is a [smooth manifold](/source/Smooth_manifold), then the [smooth functions](/source/Smooth_function) from *X* to the [real numbers](/source/Real_number) form a ring *C*∞(*X*). The set of all smooth [vector fields](/source/Vector_field) defined on *X* forms a module over *C*∞(*X*), and so do the [tensor fields](/source/Tensor_field) and the [differential forms](/source/Differential_form) on *X*. More generally, the sections of any [vector bundle](/source/Vector_bundle) form a [projective module](/source/Projective_module) over *C*∞(*X*), and by [Swan's theorem](/source/Swan's_theorem), every projective module is isomorphic to the module of sections of some vector bundle; the [category](/source/Category_(mathematics)) of *C*∞(*X*)-modules and the category of vector bundles over *X* are [equivalent](/source/Equivalence_of_categories).

- If *R* is any ring and *I* is any [left ideal](/source/Ring_ideal) in *R*, then *I* is a left *R*-module, and analogously right ideals in *R* are right *R*-modules.

- If *R* is a ring, we can define the [opposite ring](/source/Opposite_ring) *R*op, which has the same [underlying set](/source/Underlying_set) and the same addition operation, but the opposite multiplication: if *ab* = *c* in *R*, then *ba* = *c* in *R*op. Any *left* *R*-module *M* can then be seen to be a *right* module over *R*op, and any right module over *R* can be considered a left module over *R*op.

- [Modules over a Lie algebra](/source/Glossary_of_Lie_algebras#Representation_theory) are (associative algebra) modules over its [universal enveloping algebra](/source/Universal_enveloping_algebra).

- If *R* and *S* are rings with a [ring homomorphism](/source/Ring_homomorphism) *φ* : *R* → *S*, then every *S*-module *M* is an *R*-module by defining *rm* = *φ*(*r*)*m*. In particular, *S* itself is such an *R*-module.

## Submodules and homomorphisms

Suppose *M* is a left *R*-module and *N* is a [subgroup](/source/Subgroup) of *M*. Then *N* is a **submodule** (or more explicitly an *R*-submodule) if for any *n* in *N* and any *r* in *R*, the product *r* ⋅ *n* (or *n* ⋅ *r* for a right *R*-module) is in *N*.

If *X* is any [subset](/source/Subset) of an *R*-module *M*, then the submodule spanned by *X* is defined to be ⟨ X ⟩ = ⋂ N ⊇ X N {\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N} where *N* runs over the submodules of *M* that contain *X*, or explicitly { ∑ i = 1 k r i x i | r i ∈ R , x i ∈ X } {\textstyle {\bigl \{}\!\sum _{i=1}^{k}r_{i}x_{i}\mathrel {\big |} r_{i}\in R,\,x_{i}\in X{\bigr \}}} , which is important in the definition of [tensor products of modules](/source/Tensor_product_of_modules).[3]

The set of submodules of a given module *M*, together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a [lattice](/source/Lattice_(order)) that satisfies the **[modular law](/source/Modular_lattice)**: Given submodules *U*, *N*1, *N*2 of *M* such that *N*1 ⊆ *N*2, then the following two submodules are equal: (*N*1 + *U*) ∩ *N*2 = *N*1 + (*U* ∩ *N*2).

If *M* and *N* are left *R*-modules, then a [map](/source/Map_(mathematics)) *f* : *M* → *N* is a **[homomorphism of *R*-modules](/source/Module_homomorphism)** if for any *m*, *n* in *M* and *r*, *s* in *R*,

- f ( r ⋅ m + s ⋅ n ) = r ⋅ f ( m ) + s ⋅ f ( n ) {\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)} .

This, like any [homomorphism](/source/Homomorphism) of mathematical objects, is just a mapping that preserves the structure of the objects. Another name for a homomorphism of *R*-modules is an *R*-[linear map](/source/Linear_map).

A [bijective](/source/Bijective) module homomorphism *f* : *M* → *N* is called a module [isomorphism](/source/Isomorphism), and the two modules *M* and *N* are called **isomorphic**. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.

The [kernel](/source/Kernel_(algebra)) of a module homomorphism *f* : *M* → *N* is the submodule of *M* consisting of all elements that are sent to zero by *f*, and the [image](/source/Image_(mathematics)) of *f* is the submodule of *N* consisting of values *f*(*m*) for all elements *m* of *M*.[4] The [isomorphism theorems](/source/Isomorphism_theorem) familiar from groups and vector spaces are also valid for *R*-modules.

Given a ring *R*, the set of all left *R*-modules together with their module homomorphisms forms an [abelian category](/source/Abelian_category), denoted by *R*-**Mod** (see [category of modules](/source/Category_of_modules)).

## Types of modules

See also: [Glossary of module theory](/source/Glossary_of_module_theory)

**Finitely generated**
- An *R*-module *M* is [finitely generated](/source/Finitely_generated_module) if there exist finitely many elements *x*1, ..., *x**n* in *M* such that every element of *M* is a [linear combination](/source/Linear_combination) of those elements with coefficients from the ring *R*.

**Cyclic**
- A module is called a [cyclic module](/source/Cyclic_module) if it is generated by one element.

**Free**
- A [free *R*-module](/source/Free_module) is a module that has a basis, or equivalently, one that is isomorphic to a [direct sum](/source/Direct_sum_of_modules) of copies of the ring *R*. These are the modules that behave very much like vector spaces.

**Projective**
- [Projective modules](/source/Projective_module) are [direct summands](/source/Direct_summand) of free modules and share many of their desirable properties.

**Injective**
- [Injective modules](/source/Injective_module) are defined dually to projective modules.

**Flat**
- A module is called [flat](/source/Flat_module) if taking the [tensor product](/source/Tensor_product_of_modules) of it with any [exact sequence](/source/Exact_sequence) of *R*-modules preserves exactness.

**Torsionless**
- A module is called [torsionless](/source/Torsionless_module) if it embeds into its [algebraic dual](/source/Dual_module).

**Simple**
- A [simple module](/source/Simple_module) *S* is a module that is not {0} and whose only submodules are {0} and *S*. Simple modules are sometimes called *irreducible*.[5]

**Semisimple**
- A [semisimple module](/source/Semisimple_module) is a direct sum (finite or not) of simple modules. Historically these modules are also called *completely reducible*.

**Indecomposable**
- An [indecomposable module](/source/Indecomposable_module) is a non-zero module that cannot be written as a [direct sum](/source/Direct_sum_of_modules) of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules that are not simple (e.g. [uniform modules](/source/Uniform_module)).

**Faithful**
- A [faithful module](/source/Faithful_module) *M* is one where the action of each *r* ≠ 0 in *R* on *M* is nontrivial (i.e. *r* ⋅ *x* ≠ 0 for some *x* in *M*). Equivalently, the [annihilator](/source/Annihilator_(ring_theory)) of *M* is the [zero ideal](/source/Zero_ideal).

**Torsion-free**
- A [torsion-free module](/source/Torsion-free_module) is a module over a ring such that 0 is the only element annihilated by a regular element (non [zero-divisor](/source/Zero-divisor)) of the ring, equivalently *rm* = 0 implies *r* = 0 or *m* = 0.

**Noetherian**
- A [Noetherian module](/source/Noetherian_module) is a module that satisfies the [ascending chain condition](/source/Ascending_chain_condition) on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.

**Artinian**
- An [Artinian module](/source/Artinian_module) is a module that satisfies the [descending chain condition](/source/Descending_chain_condition) on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.

**Graded**
- A [graded module](/source/Graded_module) is a module over a [graded ring](/source/Graded_ring) *R* = ⨁*x* *R**x* together with a [direct sum](/source/Direct_sum) decomposition *M* = ⨁*x* *M**x* such that *R**x**M**y* ⊆ *M**x*+*y* for all *x* and *y*.

**Uniform**
- A [uniform module](/source/Uniform_module) is a module in which all pairs of nonzero submodules have nonzero intersection.

## Further notions

### Relation to representation theory

For broader coverage of this topic, see [Representation theory](/source/Representation_theory).

A representation of a group *G* over a field *k* is a module over the [group ring](/source/Group_ring) *k*[*G*].

If *M* is a left *R*-module, then the *action* of an element *r* in *R* is defined to be the map *M* → *M* that sends each *x* to *rx* (or *xr* in the case of a right module), and is necessarily a [group endomorphism](/source/Group_homomorphism) of the abelian group (*M*, +). The set of all group endomorphisms of *M* is denoted End**Z**(*M*) and forms a ring under addition and [composition](/source/Function_composition), and sending a ring element *r* of *R* to its action actually defines a [ring homomorphism](/source/Ring_homomorphism) from *R* to End**Z**(*M*).

Such a ring homomorphism *R* → End**Z**(*M*) is called a *representation* of the abelian group *M* over the ring *R*; an alternative and equivalent way of defining left *R*-modules is to say that a left *R*-module is an abelian group *M* together with a representation of *M* over *R*. Such a representation *R* → End**Z**(*M*) may also be called a *ring action* of *R* on *M*.

A representation is called *faithful* if the map *R* → End**Z**(*M*) is [injective](/source/Injective). In terms of modules, this means that if *r* is an element of *R* such that *rx* = 0 for all *x* in *M*, then *r* = 0. Every abelian group is a faithful module over the [integers](/source/Integer) or over the [ring of integers modulo *n*](/source/Modular_arithmetic), **Z**/*n***Z**, for some *n*.

### Generalizations

A ring *R* corresponds to a [preadditive category](/source/Preadditive_category) **R** with a single [object](/source/Object_(category_theory)). With this understanding, a left *R*-module is just a covariant [additive functor](/source/Additive_functor) from **R** to the [category of abelian groups](/source/Category_of_abelian_groups) **Ab**, and right *R*-modules are contravariant additive functors. This suggests that, if **C** is any preadditive category, a covariant additive functor from **C** to **Ab** should be considered a generalized left module over **C**. These functors form a [functor category](/source/Functor_category) **C**-**Mod**, which is the natural generalization of the module category *R*-**Mod**.

Modules over *commutative* rings can be generalized in a different direction: take a [ringed space](/source/Ringed_space) (*X*, O*X*) and consider the [sheaves](/source/Sheaf_(mathematics)) of O*X*-modules (see [sheaf of modules](/source/Sheaf_of_modules)). These form a category O*X*-**Mod**, and play an important role in modern [algebraic geometry](/source/Algebraic_geometry). If *X* has only a single point, then this is a module category in the old sense over the commutative ring O*X*(*X*).

One can also consider modules over a [semiring](/source/Semiring), called a [semimodule](/source/Semimodule). Modules over rings are abelian groups, but modules over semirings are only [commutative](/source/Commutative) [monoids](/source/Monoid). Most applications of modules are still possible. In particular, for any [semiring](/source/Semiring) *S*, the matrices over *S* form a semiring over which the tuples of elements from *S* are a module (in this generalized sense only). This allows a further generalization of the concept of [vector space](/source/Vector_space) incorporating the semirings from theoretical computer science.

Over [near-rings](/source/Near-rings), one can consider near-ring modules, a nonabelian generalization of modules.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

## See also

- [G-module](/source/G-module)

- [Group ring](/source/Group_ring)

- [Algebra (ring theory)](/source/Algebra_(ring_theory))

- [Module (model theory)](/source/Module_(model_theory))

- [Module spectrum](/source/Module_spectrum)

- [Annihilator](/source/Annihilator_(ring_theory))

## Notes

1. **[^](#cite_ref-1)** Hungerford (1974) *Algebra*, Springer, p 169: "Modules over a ring are a generalization of abelian groups (which are modules over Z)."

1. **[^](#cite_ref-DummitFoote_2-0)** Dummit, David S. & Foote, Richard M. (2004). *Abstract Algebra*. Hoboken, NJ: John Wiley & Sons, Inc. [ISBN](/source/ISBN_(identifier)) [978-0-471-43334-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-43334-7).

1. **[^](#cite_ref-3)** Mcgerty, Kevin (2016). ["ALGEBRA II: RINGS AND MODULES"](http://people.maths.ox.ac.uk/mcgerty/Algebra%20II.pdf) (PDF).

1. **[^](#cite_ref-4)** Ash, Robert. ["Module Fundamentals"](https://faculty.math.illinois.edu/~r-ash/Algebra/Chapter4.pdf) (PDF). *Abstract Algebra: The Basic Graduate Year*.

1. **[^](#cite_ref-5)** Jacobson (1964), [p. 4](https://books.google.com/books?id=KlMDjaJxZAkC&pg=PA4), Def. 1

## References

- F.W. Anderson and K.R. Fuller: *Rings and Categories of Modules*, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, [ISBN](/source/ISBN_(identifier)) [0-387-97845-3](https://en.wikipedia.org/wiki/Special:BookSources/0-387-97845-3), [ISBN](/source/ISBN_(identifier)) [3-540-97845-3](https://en.wikipedia.org/wiki/Special:BookSources/3-540-97845-3)

- [Nathan Jacobson](/source/Nathan_Jacobson). *Structure of rings*. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, [ISBN](/source/ISBN_(identifier)) [978-0-8218-1037-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-1037-8)

## External links

- ["Module"](https://www.encyclopediaofmath.org/index.php?title=Module), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society), 2001 [1994]

- [module](https://ncatlab.org/nlab/show/module) at the [*n*Lab](/source/NLab)

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