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
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
. Complementarily,
noncommutative algebra 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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
equipped with two
binary operations, 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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under addition as well as a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
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
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
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 number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s form fields.
If ''
'' is a given commutative ring, then the set of all
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s in the variable
whose coefficients are in ''
'' forms the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
, denoted
. The same holds true for several variables.
If ''
'' is some
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, 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 val ...
s on ''
'' form a commutative ring. The same is true for
differentiable or
holomorphic functions, when the two concepts are defined, such as for ''
'' a
complex manifold.
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 divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s, 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
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cl ...
.
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\c ...
, 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 space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, 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, and a submodule of a free module needs not to be free.
A
module of finite type 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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
. 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
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
. 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 domains.
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 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
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
.
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
''
'' / ''
'' 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, allows the extension of certain theorems to non-Noetherian rings.
Artinian rings
A ring is called
Artinian (after
Emil Artin), 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 In the branch of abstract algebra called ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring ''R'' (with 1) is called semiprimary ...
, 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