TheInfoList

OR:

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 Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
in which the multiplication operation 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. Most familiar as the name of ...
. 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. Promin ...
. 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 ...
$R$ 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 ...
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 "$\cdot$"; e.g. $a+b$ and $a \cdot b$. 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 commu ...
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. Monoids ...
under multiplication, where multiplication distributes over addition; i.e., $a \cdot \left\left(b + c\right\right) = \left\left(a \cdot b\right\right) + \left\left(a \cdot c\right\right)$. The identity elements for addition and multiplication are denoted $0$ and $1$, respectively. If the multiplication is commutative, i.e. $a \cdot b = b \cdot a,$ then the ring ''$R$'' 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 $\mathbb$ 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 $\mathbb$ 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 $0 \not = 1$ and every non-zero element $a$ is invertible; i.e., has a multiplicative inverse $b$ such that $a \cdot b = 1$. Therefore, by definition, any field is a commutative ring. The
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
, 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 ''$R$'' 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 ex ...
s in the variable $X$ whose coefficients are in ''$R$'' 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 variabl ...
, denoted . The same holds true for several variables. If ''$V$'' 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 poi ...
, for example a subset of some $\mathbb^n$, 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 va ...
s on ''$V$'' form a commutative ring. The same is true for
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
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 der ...
s, when the two concepts are defined, such as for ''$V$'' 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 ...
.

# Divisibility

In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element $a$ of ring ''$R$'' is called a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
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 $a$ such that there exists a non-zero element $b$ of the ring such that $ab = 0$. If ''$R$'' possesses no non-zero zero divisors, it is called an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
(or domain). An element $a$ satisfying $a^n = 0$ for some positive integer $n$ 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 cla ...
.

## 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 ''$S$'' is a
multiplicatively closed subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...
of ''$R$'' (i.e. whenever $s,t \in S$ then so is $st$) then the ''localization'' of ''$R$'' at ''$S$'', or ''ring of fractions'' with denominators in ''$S$'', usually denoted $S^R$ consists of symbols subject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language ''$\mathbb$'' is the localization of ''$\mathbb$'' at all nonzero integers. This construction works for any integral domain ''$R$'' instead of ''$\mathbb$''. The localization $\left\left(R\backslash \left\\right\right)^R$ is a field, called the
quotient field In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of ''$R$''.

# 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, which simplifies the situation considerably.

## Modules

For a ring ''$R$'', an ''$R$''-''module'' ''$M$'' 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 ''$R$'' 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 ...
s, since there are modules that do not have any
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
, 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 fie ...
, 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. 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 ''$R$'' are the
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the m ...
s of ''$R$'', i.e., the modules contained in ''$R$''. In more detail, an ideal ''$I$'' is a non-empty subset of ''$R$'' such that for all ''$r$'' in ''$R$'', ''$i$'' and ''$j$'' in ''$I$'', both ''$ri$'' and ''$i+j$'' are in ''$I$''. 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 ''$\left\$'' and ''$R$'', the whole ring. These two ideals are the only ones precisely if ''$R$'' is a field. Given any subset ''$F=\left\_$'' of ''$R$'' (where ''$J$'' is some index set), the ideal ''generated by $F$'' is the smallest ideal that contains ''$F$''. Equivalently, it is given by finite linear combinations ''$r_1 f_1 + r_2 f_2 + \dots + r_n f_n .$''

### Principal ideal domains

If ''$F$'' consists of a single element ''$r$'', the ideal generated by ''$F$'' consists of the multiples of ''$r$'', i.e., the elements of the form ''$rs$'' for arbitrary elements ''$s$''. 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, ''$R$'' is called a principal ideal ring; two important cases are ''$\mathbb$'' and '''', the polynomial ring over a field ''$k$''. 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 princip ...
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 ''$R$'' is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
(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 ''$a=bc ,$'' is by either ''$b$'' or ''$c$'' being a unit. An example, important in field theory, are
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
s, i.e., irreducible elements in '''', for a field ''$k$''. The fact that ''$\mathbb$'' 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 In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the o ...
. An element ''$a$'' 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 ''$a$'' divides a product ''$bc$'', ''$a$'' divides ''$b$'' or ''$c$''. 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" ''$I$'' "out" gives another ring, the ''factor ring'' ''$R$'' / ''$I$'': it is the set of
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and righ ...
s of ''$I$'' together with the operations ''$\left(a+I\right)+\left(b+I\right)=\left(a+b\right)+I$'' and ''$\left\left(a+I\right\right) \left\left(b+I\right\right)=ab+I$''. For example, the ring $\mathbb/n\mathbb$ (also denoted $\mathbb_n$), where ''$n$'' is an integer, is the ring of integers modulo ''$n$''. 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 b ...
. 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 ''$m$'' 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 bico ...
''$R$'' / ''$m$'' is a field. Except for the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
, 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 ''$0 \subseteq I_0 \subseteq I_1 \subseteq \dots \subseteq I_n \subseteq I_ \dots$'' becomes stationary, i.e. becomes constant beyond some index ''$n$''. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent,
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the m ...
s 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 ''$R$'' is Noetherian, then so is the polynomial ring '''' (by Hilbert's basis theorem), any localization ''$S^R$'', and also any factor ring ''$R$'' / ''$I$''. Any non-Noetherian ring ''$R$'' 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 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 ...
), if every descending chain of ideals ''$R \supseteq I_0 \supseteq I_1 \supseteq \dots \supseteq I_n \supseteq I_ \dots$'' becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, ''$\mathbb$'' is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain ''$\mathbb \supsetneq 2\mathbb \supsetneq 4\mathbb \supsetneq 8\mathbb \dots$'' 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, $\mathbb$ is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
. This is not true for more general rings, as algebraists realized in the 19th century. For example, in
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
$R \setminus p$ is multiplicatively closed. The localisation $\left\left(R \setminus p\right\right)^R$ is important enough to have its own notation: $R_p$. This ring has only one maximal ideal, namely $pR_p$. Such rings are called
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administra ...
.

## The spectrum

The ''spectrum of a ring $R$'',This notion can be related to the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of a linear operator, see Spectrum of a C*-algebra and Gelfand representation.
denoted by ''$\text\ R$'', is the set of all prime ideals of ''$R$''. It is equipped with a topology, the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, which reflects the algebraic properties of ''$R$'': a basis of open subsets is given by ''$D\left(f\right) = \left\$'', where ''$f$'' is any ring element. Interpreting ''$f$'' as a function that takes the value ''f'' mod ''p'' (i.e., the image of ''f'' in the residue field ''R''/''p''), this subset is the locus where ''f'' is non-zero. The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps ''R'' → ''R''''f'' and ''R'' → ''R'' / ''fR'' correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
and closed immersions respectively. Even for basic rings, such as illustrated for ''R'' = Z at the right, the Zariski topology is quite different from the one on the set of real numbers. The spectrum contains the set of maximal ideals, which is occasionally denoted mSpec (''R''). For an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
''k'', mSpec (k 'T''1, ..., ''T''''n''/ (''f''1, ..., ''f''''m'')) is in bijection with the set Thus, maximal ideals reflect the geometric properties of solution sets of polynomials, which is an initial motivation for the study of commutative rings. However, the consideration of non-maximal ideals as part of the geometric properties of a ring is useful for several reasons. For example, the minimal prime ideals (i.e., the ones not strictly containing smaller ones) correspond to the
irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal ( ...
s of Spec ''R''. For a Noetherian ring ''R'', Spec ''R'' has only finitely many irreducible components. This is a geometric restatement of primary decomposition, according to which any ideal can be decomposed as a product of finitely many primary ideals. This fact is the ultimate generalization of the decomposition into prime ideals in Dedekind rings.

## Affine schemes

The notion of a spectrum is the common basis of commutative algebra and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometric ...
. Algebraic geometry proceeds by endowing Spec ''R'' with a sheaf $\mathcal O$ (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an
affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
. Given an affine scheme, the underlying ring ''R'' can be recovered as the global sections of $\mathcal O$. Moreover, this one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any ''f'' : ''R'' → ''S'' gives rise to a
continuous map 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 valu ...
in the opposite direction The resulting equivalence of the two said categories aptly reflects algebraic properties of rings in a geometrical manner. Similar to the fact that
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
are locally given by open subsets of R''n'', affine schemes are local models for schemes, which are the object of study in algebraic geometry. Therefore, several notions concerning commutative rings stem from geometric intuition.

## Dimension

The ''Krull dimension'' (or dimension) dim ''R'' of a ring ''R'' measures the "size" of a ring by, roughly speaking, counting independent elements in ''R''. The dimension of algebras over a field ''k'' can be axiomatized by four properties: * The dimension is a local property: dim ''R'' = supp ∊ Spec ''R'' dim ''R''''p''. * The dimension is independent of nilpotent elements: if ''I'' ⊆ ''R'' is nilpotent then dim ''R'' = dim ''R'' / ''I''. * The dimension remains constant under a finite extension: if ''S'' is an ''R''-algebra which is finitely generated as an ''R''-module, then dim ''S'' = dim ''R''. * The dimension is calibrated by dim ''k'' 'X''1, ..., ''X''''n''= ''n''. This axiom is motivated by regarding the polynomial ring in ''n'' variables as an algebraic analogue of ''n''-dimensional space. The dimension is defined, for any ring ''R'', as the supremum of lengths ''n'' of chains of prime ideals For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. The integers are one-dimensional, since chains are of the form (0) ⊊ (''p''), where ''p'' is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
. For non-Noetherian rings, and also non-local rings, the dimension may be infinite, but Noetherian local rings have finite dimension. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in
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. Promin ...
, the going-up theorem and Krull's principal ideal theorem.

# Ring homomorphisms

A ''ring homomorphism'' or, more colloquially, simply a ''map'', is a map ''f'' : ''R'' → ''S'' such that These conditions ensure ''f''(0) = 0. Similarly as for other algebraic structures, a ring homomorphism is thus a map that is compatible with the structure of the algebraic objects in question. In such a situation ''S'' is also called an ''R''-algebra, by understanding that ''s'' in ''S'' may be multiplied by some ''r'' of ''R'', by setting The ''kernel'' and ''image'' of ''f'' are defined by ker (''f'') = and im (''f'') = ''f''(''R'') = . The kernel is an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of ''R'', and the image is a
subring In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of ''S''. A ring homomorphism is called an isomorphism if it is bijective. An example of a ring isomorphism, known as the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
, is $\mathbf Z/n = \bigoplus_^k \mathbf Z/p_i$ where ''n'' = ''p''1''p''2...''p''''k'' is a product of pairwise distinct
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
s. Commutative rings, together with ring homomorphisms, form a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
. The ring Z is the
initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element ...
in this category, which means that for any commutative ring ''R'', there is a unique ring homomorphism Z → ''R''. By means of this map, an integer ''n'' can be regarded as an element of ''R''. For example, the
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
$(a+b)^n = \sum_^n \binom n k a^k b^$ which is valid for any two elements ''a'' and ''b'' in any commutative ring ''R'' is understood in this sense by interpreting the binomial coefficients as elements of ''R'' using this map. Given two ''R''-algebras ''S'' and ''T'', their
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
is again a commutative ''R''-algebra. In some cases, the tensor product can serve to find a ''T''-algebra which relates to ''Z'' as ''S'' relates to ''R''. For example,

## Finite generation

An ''R''-algebra ''S'' is called finitely generated (as an algebra) if there are finitely many elements ''s''1, ..., ''s''''n'' such that any element of ''s'' is expressible as a polynomial in the ''s''''i''. Equivalently, ''S'' is isomorphic to A much stronger condition is that ''S'' is finitely generated as an ''R''-module, which means that any ''s'' can be expressed as a ''R''-linear combination of some finite set ''s''1, ..., ''s''''n''.

# Local rings

A ring is called
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administra ...
if it has only a single maximal ideal, denoted by ''m''. For any (not necessarily local) ring ''R'', the localization at a prime ideal ''p'' is local. This localization reflects the geometric properties of Spec ''R'' "around ''p''". Several notions and problems in commutative algebra can be reduced to the case when ''R'' is local, making local rings a particularly deeply studied class of rings. The residue field of ''R'' is defined as Any ''R''-module ''M'' yields a ''k''-vector space given by ''M'' / ''mM''. Nakayama's lemma shows this passage is preserving important information: a finitely generated module ''M'' is zero if and only if ''M'' / ''mM'' is zero.

## Regular local rings

The ''k''-vector space ''m''/''m''2 is an algebraic incarnation of the
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
. Informally, the elements of ''m'' can be thought of as functions which vanish at the point ''p'', whereas ''m''2 contains the ones which vanish with order at least 2. For any Noetherian local ring ''R'', the inequality holds true, reflecting the idea that the cotangent (or equivalently the tangent) space has at least the dimension of the space Spec ''R''. If equality holds true in this estimate, ''R'' is called a
regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ...
. A Noetherian local ring is regular if and only if the ring (which is the ring of functions on the tangent cone) $\bigoplus_n m^n / m^$ is isomorphic to a polynomial ring over ''k''. Broadly speaking, regular local rings are somewhat similar to polynomial rings. Regular local rings are UFD's.
Discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...
s are equipped with a function which assign an integer to any element ''r''. This number, called the valuation of ''r'' can be informally thought of as a zero or pole order of ''r''. Discrete valuation rings are precisely the one-dimensional regular local rings. For example, the ring of germs of holomorphic functions on a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
is a discrete valuation ring.

## Complete intersections

By Krull's principal ideal theorem, a foundational result in the dimension theory of rings, the dimension of is at least ''r'' − ''n''. A ring ''R'' is called a complete intersection ring if it can be presented in a way that attains this minimal bound. This notion is also mostly studied for local rings. Any regular local ring is a complete intersection ring, but not conversely. A ring ''R'' is a ''set-theoretic'' complete intersection if the reduced ring associated to ''R'', i.e., the one obtained by dividing out all nilpotent elements, is a complete intersection. As of 2017, it is in general unknown, whether curves in three-dimensional space are set-theoretic complete intersections.

## Cohen–Macaulay rings

The depth of a local ring ''R'' is the number of elements in some (or, as can be shown, any) maximal regular sequence, i.e., a sequence ''a''1, ..., ''a''''n'' ∈ ''m'' such that all ''a''''i'' are non-zero divisors in For any local Noetherian ring, the inequality holds. A local ring in which equality takes place is called a
Cohen–Macaulay ring In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a fi ...
. Local complete intersection rings, and a fortiori, regular local rings are Cohen–Macaulay, but not conversely. Cohen–Macaulay combine desirable properties of regular rings (such as the property of being universally catenary rings, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.

# Constructing commutative rings

There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable. For example, an integral domain that is integrally closed in its
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fiel ...
is called normal. This is a desirable property, for example any normal one-dimensional ring is necessarily regular. Rendering a ring normal is known as ''normalization''.

## Completions

If ''I'' is an ideal in a commutative ring ''R'', the powers of ''I'' form topological neighborhoods of ''0'' which allow ''R'' to be viewed as a topological ring. This topology is called the ''I''-adic topology. ''R'' can then be completed with respect to this topology. Formally, the ''I''-adic completion is the
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
of the rings ''R''/''In''. For example, if ''k'' is a field, ''k'' ''X'', the
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
ring in one variable over ''k'', is the ''I''-adic completion of ''k'' 'X''where ''I'' is the principal ideal generated by ''X''. This ring serves as an algebraic analogue of the disk. Analogously, the ring of ''p''-adic integers is the completion of Z with respect to the principal ideal (''p''). Any ring that is isomorphic to its own completion, is called complete. Complete local rings satisfy Hensel's lemma, which roughly speaking allows extending solutions (of various problems) over the residue field ''k'' to ''R''.

# Homological notions

Several deeper aspects of commutative rings have been studied using methods from
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topo ...
. lists some open questions in this area of active research.

## Projective modules and Ext functors

Projective modules can be defined to be the
direct summand 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. To see how the direct sum is used in abstract algebra, consider a mo ...
s of free modules. If ''R'' is local, any finitely generated projective module is actually free, which gives content to an analogy between projective modules and
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 every p ...
s. The Quillen–Suslin theorem asserts that any finitely generated projective module over ''k'' 'T''1, ..., ''T''''n''(''k'' a field) is free, but in general these two concepts differ. A local Noetherian ring is regular if and only if its
global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
is finite, say ''n'', which means that any finitely generated ''R''-module has a
resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual ma ...
by projective modules of length at most ''n''. The proof of this and other related statements relies on the usage of homological methods, such as the
Ext functor In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic str ...
. This functor is the derived functor of the functor The latter functor is exact if ''M'' is projective, but not otherwise: for a surjective map ''E'' → ''F'' of ''R''-modules, a map ''M'' → ''F'' need not extend to a map ''M'' → ''E''. The higher Ext functors measure the non-exactness of the Hom-functor. The importance of this standard construction in homological algebra stems can be seen from the fact that a local Noetherian ring ''R'' with residue field ''k'' is regular if and only if vanishes for all large enough ''n''. Moreover, the dimensions of these Ext-groups, known as Betti numbers, grow polynomially in ''n'' if and only if ''R'' is a local complete intersection ring. A key argument in such considerations is the Koszul complex, which provides an explicit free resolution of the residue field ''k'' of a local ring ''R'' in terms of a regular sequence.

## Flatness

The
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
is another non-exact functor relevant in the context of commutative rings: for a general ''R''-module ''M'', the functor is only right exact. If it is exact, ''M'' is called flat. If ''R'' is local, any finitely presented flat module is free of finite rank, thus projective. Despite being defined in terms of homological algebra, flatness has profound geometric implications. For example, if an ''R''-algebra ''S'' is flat, the dimensions of the fibers (for prime ideals ''p'' in ''R'') have the "expected" dimension, namely dim ''S'' − dim ''R'' + dim (''R'' / ''p'').

# Properties

By Wedderburn's theorem, every finite
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
is commutative, and therefore a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, sub ...
. Another condition ensuring commutativity of a ring, due to Jacobson, is the following: for every element ''r'' of ''R'' there exists an integer such that . If, ''r''2 = ''r'' for every ''r'', the ring is called
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...
. More general conditions which guarantee commutativity of a ring are also known.

# Generalizations

A graded ring ''R'' = ⨁''i''∊Z ''R''''i'' is called graded-commutative if, for all homogeneous elements ''a'' and ''b'', If the ''R''''i'' are connected by differentials ∂ such that an abstract form of the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
holds, i.e., ''R'' is called a commutative differential graded algebra (cdga). An example is the complex of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
, with the multiplication given by the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of t ...
, is a cdga. The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the cohomology ring. A broad range examples of graded rings arises in this way. For example, the Lazard ring is the ring of cobordism classes of complex manifolds. A graded-commutative ring with respect to a grading by Z/2 (as opposed to Z) is called a
superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. T ...
. A related notion is an almost commutative ring, which means that ''R'' is
filtered Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
in such a way that the associated graded ring is commutative. An example is the Weyl algebra and more general rings of
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s.

## Simplicial commutative rings

A simplicial commutative ring is a simplicial object in the category of commutative rings. They are building blocks for (connective) derived algebraic geometry. A closely related but more general notion is that of E-ring.

# Applications of the commutative rings

*
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 der ...
s * Algebraic K-theory * Topological K-theory * Divided power structures * Witt vectors * Hecke algebra (used in Wiles's proof of Fermat's Last Theorem) * Fontaine's period rings * Cluster algebra * Convolution algebra (of a commutative group) * Fréchet algebra

* Almost ring, a certain generalization of a commutative ring *
Divisibility (ring theory) In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. ...
:
nilpotent element 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 cla ...
, (ex.
dual number In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Du ...
s) * Ideals and modules: Radical of an ideal,
Morita equivalence In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of module ...
*
Ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
s:
integral element In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...
:
Cayley–Hamilton theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfie ...
,
Integrally closed domain In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ...
, Krull ring, Krull–Akizuki theorem, Mori–Nagata theorem * Primes: Prime avoidance lemma, Jacobson radical, Nilradical of a ring, Spectrum:
Compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
, Connected ring, Differential calculus over commutative algebras, Banach–Stone theorem *
Local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
s: Gorenstein local ring (also used in Wiles's proof of Fermat's Last Theorem):
Duality (mathematics) In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
,
Eben Matlis Eben Matlis (August 28, 1923 - March 27, 2015) was a mathematician known for his contributions to the theory of rings and modules, especially for his work with injective modules over commutative Noetherian rings, and his introduction of Matlis dua ...
; Dualizing module, Popescu's theorem, Artin approximation theorem.

* * * * * * *