In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a commutative ring is a
ring in which the multiplication operation is
commutative. The study of commutative rings is called
commutative algebra. Complementarily,
noncommutative algebra
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a ...
is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
Definition and first examples
Definition
A ''ring'' is a
set equipped with two
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "
" and "
"; e.g.
and
. To form a ring these two operations have to satisfy a number of properties: the ring has to be an
abelian group under addition as well as a
monoid under multiplication, where multiplication
distributes over addition; i.e.,
. The identity elements for addition and multiplication are denoted
and
, respectively.
If the multiplication is commutative, i.e.
then the ring ''
'' is called ''commutative''. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.
First examples
An important example, and in some sense crucial, is the
ring of integers
with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted
as an abbreviation of the
German word ''Zahlen'' (numbers).
A
field is a commutative ring where
and every
non-zero element
is invertible; i.e., has a multiplicative inverse
such that
. Therefore, by definition, any field is a commutative ring. The
rational,
real and
complex numbers form fields.
If ''
'' is a given commutative ring, then the set of all
polynomials in the variable
whose coefficients are in ''
'' forms the
polynomial ring, denoted
. The same holds true for several variables.
If ''
'' is some
topological space, for example a subset of some
, real- or complex-valued
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s on ''
'' form a commutative ring. The same is true for
differentiable or
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s, when the two concepts are defined, such as for ''
'' a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
.
Divisibility
In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of
divisibility for rings is richer. An element
of ring ''
'' is called a
unit if it possesses a multiplicative inverse. Another particular type of element is the
zero divisors, i.e. an element
such that there exists a non-zero element
of the ring such that
. If ''
'' possesses no non-zero zero divisors, it is called an
integral domain (or domain). An element
satisfying
for some positive integer
is called
nilpotent.
Localizations
The ''localization'' of a ring is a process in which some elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if ''
'' is a
multiplicatively closed subset of ''
'' (i.e. whenever
then so is
) then the ''localization'' of ''
'' at ''
'', or ''ring of fractions'' with denominators in ''
'', usually denoted
consists of symbols
subject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language ''
'' is the localization of ''
'' at all nonzero integers. This construction works for any integral domain ''
'' instead of ''
''. The localization
is a field, called the
quotient field of ''
''.
Ideals and modules
Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically
two-sided
In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F of a manifold M is said to be 2-sided in M when there is an embedding
::h\colon F\times 1,1to M
with h(x,0)=x for each x\in F and
::h(F\times 1,1\ ...
, which simplifies the situation considerably.
Modules
For a ring ''
'', an ''
''-''module'' ''
'' is like what a vector space is to a field. That is, elements in a module can be added; they can be multiplied by elements of ''
'' subject to the same axioms as for a vector space.
The study of modules is significantly more involved than the one of
vector spaces, since there are modules that do not have any
basis, that is, do not contain a
spanning set whose elements are
linearly independents. A module that has a basis is called a
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
, and a submodule of a free module needs not to be free.
A
module of finite type
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts inclu ...
is a module that has a finite spanning set. Modules of finite type play a fundamental role in the theory of commutative rings, similar to the role of the
finite-dimensional vector spaces in
linear algebra. In particular,
Noetherian rings (see also , below) can be defined as the rings such that every submodule of a module of finite type is also of finite type.
Ideals
''Ideals'' of a ring ''
'' are the
submodules of ''
'', i.e., the modules contained in ''
''. In more detail, an ideal ''
'' is a non-empty subset of ''
'' such that for all ''
'' in ''
'', ''
'' and ''
'' in ''
'', both ''
'' and ''
'' are in ''
''. For various applications, understanding the ideals of a ring is of particular importance, but often one proceeds by studying modules in general.
Any ring has two ideals, namely the
zero ideal ''
'' and ''
'', the whole ring. These two ideals are the only ones precisely if ''
'' is a field. Given any subset ''
'' of ''
'' (where ''
'' is some index set), the ideal ''generated by
'' is the smallest ideal that contains ''
''. Equivalently, it is given by finite
linear combinations
''
''
Principal ideal domains
If ''
'' consists of a single element ''
'', the ideal generated by ''
'' consists of the multiples of ''
'', i.e., the elements of the form ''
'' for arbitrary elements ''
''. Such an ideal is called a
principal ideal. If every ideal is a principal ideal, ''
'' is called a
principal ideal ring; two important cases are ''
'' and ''
'', the polynomial ring over a field ''
''. These two are in addition domains, so they are called
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
s.
Unlike for general rings, for a principal ideal domain, the properties of individual elements are strongly tied to the properties of the ring as a whole. For example, any principal ideal domain ''
'' is a
unique factorization domain (UFD) which means that any element is a product of irreducible elements, in a (up to reordering of factors) unique way. Here, an element ''a'' in a domain is called
irreducible if the only way of expressing it as a product
''
''
is by either ''
'' or ''
'' being a unit. An example, important in
field theory, are
irreducible polynomials, i.e., irreducible elements in ''
'', for a field ''
''. The fact that ''
'' is a UFD can be stated more elementarily by saying that any natural number can be uniquely decomposed as product of powers of prime numbers. It is also known as the
fundamental theorem of arithmetic.
An element ''
'' is a
prime element
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish pri ...
if whenever ''
'' divides a product ''
'', ''
'' divides ''
'' or ''
''. In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general.
The factor ring
The definition of ideals is such that "dividing" ''
'' "out" gives another ring, the ''factor ring'' ''
'' / ''
'': it is the set of
cosets of ''
'' together with the operations
''
'' and ''
''.
For example, the ring
(also denoted
), where ''
'' is an integer, is the ring of integers modulo ''
''. It is the basis of
modular arithmetic.
An ideal is ''proper'' if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called
maximal. An ideal ''
'' is maximal
if and only if ''
'' / ''
'' is a field. Except for the
zero ring, any ring (with identity) possesses at least one maximal ideal; this follows from
Zorn's lemma.
Noetherian rings
A ring is called ''Noetherian'' (in honor of
Emmy Noether, who developed this concept) if every
ascending chain of ideals
''
''
becomes stationary, i.e. becomes constant beyond some index ''
''. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent,
submodules of finitely generated modules are finitely generated.
Being Noetherian is a highly important finiteness condition, and the condition is preserved under many operations that occur frequently in geometry. For example, if ''
'' is Noetherian, then so is the polynomial ring ''
'' (by
Hilbert's basis theorem), any localization ''
'', and also any factor ring ''
'' / ''
''.
Any non-Noetherian ring ''
'' is the
union of its Noetherian subrings. This fact, known as
Noetherian approximation In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
, allows the extension of certain theorems to non-Noetherian rings.
Artinian rings
A ring is called
Artinian (after
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
), if every descending chain of ideals
''
''
becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, ''
'' is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain
''
''
shows. In fact, by the
Hopkins–Levitzki theorem, every Artinian ring is Noetherian. More precisely, Artinian rings can be characterized as the Noetherian rings whose Krull dimension is zero.
The spectrum of a commutative ring
Prime ideals
As was mentioned above,
is a
unique factorization domain. This is not true for more general rings, as algebraists realized in the 19th century. For example, in