In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a module is a generalization of the notion of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
in which the
field of
scalars is replaced by a (not necessarily
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
)
ring. The concept of a ''module'' also generalizes the notion of an
abelian group, since the abelian groups are exactly the modules over the ring of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s.
Like a vector space, a module is an additive abelian group, and scalar multiplication is
distributive over the operations of addition between elements of the ring or module and is
compatible with the ring multiplication.
Modules are very closely related to the
representation theory of
groups. They are also one of the central notions of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
and
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, and are used widely in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
.
Introduction and definition
Motivation
In a vector space, the set of
scalars is a
field and acts on the vectors by scalar multiplication, subject to certain axioms such as the
distributive law. In a module, the scalars need only be a
ring, so the module concept represents a significant generalization. In commutative algebra, both
ideals and
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
s 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
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or n ...
" ring, such as a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a
basis, and, even for those that do (
free module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
s), 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) if the underlying ring does not satisfy the
invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
is then unique. (These last two assertions require the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
in general, but not in the case of
finite-dimensional vector spaces, or certain well-behaved infinite-dimensional vector spaces such as
L''p'' spaces.)
Formal definition
Suppose that ''R'' is a
ring, and 1 is its multiplicative identity.
A left ''R''-module ''M'' consists of an
abelian group and an operation such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have
#
,
#
,
#
,
#
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 .
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
:
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 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, all rings and modules are assumed to be unital.
An (''R'',''S'')-
bimodule
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, i ...
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 for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''.
If ''R'' is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, 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, then ''K''-modules are called ''K''-
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s (vector spaces over ''K'').
*If ''K'' is a field, and ''K''
'x''a univariate
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
, then a
''K'' module">'x''module ''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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
from ''M'' to ''M''. Applying the
structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the
rational and
Jordan canonical forms.
*The concept of a Z-module agrees with the notion of an abelian group. That is, every
abelian group is a module over the ring of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s Z in a unique way. For , let (''n'' summands), , and . Such a module need not have a
basis—groups containing
torsion elements do not. (For example, in the group of integers
modulo
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
3, one cannot find even one element that satisfies the definition of a
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...
set, since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
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 (including negative ones) form a module over the integers. Only
singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no
rank, in the usual sense of linear algebra. However this module has a
torsion-free rank equal to 1.
*If ''R'' is any ring and ''n'' a
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
, then the
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
''R''
''n'' is both a left and right ''R''-module over ''R'' if we use the component-wise operations. Hence when , ''R'' is an ''R''-module, where the scalar multiplication is just ring multiplication. The case yields the trivial ''R''-module consisting only of its identity element. Modules of this type are called
free and if ''R'' has
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
matrices over a ring ''R'', ''M'' is an M
''n''(''R'')-module, and ''e''
''i'' is the matrix with 1 in the -entry (and zeros elsewhere), then ''e''
''i''''M'' is an ''R''-module, since . So ''M'' breaks up as the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of ''R''-modules, . Conversely, given an ''R''-module ''M''
0, then ''M''
0⊕''n'' is an M
''n''(''R'')-module. In fact, the
category of ''R''-modules and the
category of M
''n''(''R'')-modules are
equivalent. 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 set, ''M'' is a left ''R''-module, and ''M''
''S'' is the collection of all
functions , then with addition and scalar multiplication in ''M''
''S'' defined pointwise by and , ''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'' (see below) is an ''R''-module (and in fact a ''submodule'' of ''N''
''M'').
*If ''X'' is a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
, then the
smooth functions from ''X'' to the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s form a ring ''C''
∞(''X''). The set of all smooth
vector fields defined on ''X'' forms a module over ''C''
∞(''X''), and so do the
tensor fields and the
differential forms on ''X''. More generally, the sections of any
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
form a
projective module over ''C''
∞(''X''), and by
Swan's theorem, every projective module is isomorphic to the module of sections of some vector bundle; the
category of ''C''
∞(''X'')-modules and the category of vector bundles over ''X'' are
equivalent.
*If ''R'' is any ring and ''I'' is any
left 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 ''R''
op, which has the same
underlying set and the same addition operation, but the opposite multiplication: if in ''R'', then 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 are (associative algebra) modules over its
universal enveloping algebra.
*If ''R'' and ''S'' are rings with a
ring homomorphism , then every ''S''-module ''M'' is an ''R''-module by defining . In particular, ''S'' itself is such an ''R''-module.
Submodules and homomorphisms
Suppose ''M'' is a left ''R''-module and ''N'' is a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
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 (or for a right ''R''-module) is in ''N''.
If ''X'' is any
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of an ''R''-module ''M'', then the submodule spanned by ''X'' is defined to be
where ''N'' runs over the submodules of ''M'' that contain ''X'', or explicitly
, which is important in the definition of
tensor products of modules.
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 that satisfies the
modular law:
Given submodules ''U'', ''N''
1, ''N''
2 of ''M'' such that , then the following two submodules are equal: .
If ''M'' and ''N'' are left ''R''-modules, then a
map is a
homomorphism of ''R''-modules if for any ''m'', ''n'' in ''M'' and ''r'', ''s'' in ''R'',
:
.
This, like any
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
.
A
bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
module homomorphism is called a module
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
, 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 of a module homomorphism is the submodule of ''M'' consisting of all elements that are sent to zero by ''f'', and the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of ''f'' is the submodule of ''N'' consisting of values ''f''(''m'') for all elements ''m'' of ''M''. The
isomorphism theorems 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
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.
The motivating prototypical example of an abelian category is the category o ...
, denoted by ''R''-Mod (see
category of modules).
Types of modules
; Finitely generated: An ''R''-module ''M'' is
finitely generated if there exist finitely many elements ''x''
1, ..., ''x''
''n'' in ''M'' such that every element of ''M'' is a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of those elements with coefficients from the ring ''R''.
; Cyclic: A module is called a
cyclic module In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z ...
if it is generated by one element.
; Free: A
free ''R''-module is a module that has a basis, or equivalently, one that is isomorphic to a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of copies of the ring ''R''. These are the modules that behave very much like vector spaces.
; Projective:
Projective modules are
direct summands of free modules and share many of their desirable properties.
; Injective:
Injective modules are defined dually to projective modules.
; Flat: A module is called
flat if taking the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of it with any
exact sequence of ''R''-modules preserves exactness.
; Torsionless: A module is called
torsionless if it embeds into its
algebraic dual.
; Simple: A
simple module ''S'' is a module that is not and whose only submodules are and ''S''. Simple modules are sometimes called ''irreducible''.
[Jacobson (1964)]
p. 4
Def. 1
; Semisimple: A
semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called ''completely reducible''.
; Indecomposable: An
indecomposable module is a non-zero module that cannot be written as a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules that are not simple (e.g.
uniform modules).
; Faithful: A
faithful module
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of .
Over an integral domain, a module that has a nonzero annihilator ...
''M'' is one where the action of each in ''R'' on ''M'' is nontrivial (i.e. for some ''x'' in ''M''). Equivalently, the
annihilator of ''M'' is the
zero ideal.
; Torsion-free: A
torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non
zero-divisor) of the ring, equivalently implies or .
; Noetherian: A
Noetherian module
In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the pr ...
is a module that satisfies the
ascending chain condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
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 Artinian may refer to:
Mathematics
*Objects named for Austrian mathematician Emil Artin (1898–1962)
**Artinian ideal, an ideal ''I'' in ''R'' for which the Krull dimension of the quotient ring ''R/I'' is 0
**Artinian ring, a ring which satisfies ...
is a module that satisfies the
descending chain condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important r ...
on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
; Graded: A
graded module is a module with a decomposition as a direct sum over a
graded ring such that for all ''x'' and ''y''.
; Uniform: A
uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.
Further notions
Relation to representation theory
A representation of a group ''G'' over a field ''k'' is a module over the
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 that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a
group endomorphism of the abelian group . The set of all group endomorphisms of ''M'' is denoted End
Z(''M'') and forms a ring under addition and
composition, and sending a ring element ''r'' of ''R'' to its action actually defines a
ring homomorphism from ''R'' to End
Z(''M'').
Such a ring homomorphism 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 may also be called a ''ring action'' of ''R'' on ''M''.
A representation is called ''faithful'' if the map is
injective. In terms of modules, this means that if ''r'' is an element of ''R'' such that for all ''x'' in ''M'', then . Every abelian group is a faithful module over the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s or over the
ring of integers modulo ''n'', Z/''n''Z, for some ''n''.
Generalizations
A ring ''R'' corresponds to a
preadditive category R with a single
object. With this understanding, a left ''R''-module is just a covariant
additive functor from R to the
category Ab of abelian groups, 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
In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object i ...
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 (''X'', O
''X'') and consider the
sheaves of O
''X''-modules (see
sheaf of modules). These form a category O
''X''-Mod, and play an important role in modern
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. 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
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ...
. Modules over rings are abelian groups, but modules over semirings are only
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
monoids. Most applications of modules are still possible. In particular, for any
semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ...
''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
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
incorporating the semirings from theoretical computer science.
Over
near-rings, one can consider near-ring modules, a nonabelian generalization of modules.
See also
*
Group ring
*
Algebra (ring theory)
*
Module (model theory)
*
Module spectrum
*
Annihilator
Notes
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, ,
*
Nathan Jacobson. ''Structure of rings''. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964,
External links
*
*
{{Authority control
Algebraic structures
* Module