Editing Commutative ring

{{short description|Algebraic structure}}
{{merge from|Composition ring| discuss=Talk:Composition ring#Merge?|date=May 2025}}
In [[mathematics]], a '''commutative ring''' is a [[Ring (mathematics)|ring]] in which the multiplication operation is [[commutative]]. The study of commutative rings is called [[commutative algebra]]. 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.

Commutative rings appear in the following chain of [[subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}

{{Ring theory sidebar}}
{{Algebraic structures |Ring}}

== Definition and first examples ==

=== Definition ===
{{details|topic=the definition of rings|Ring (mathematics)}}
A ''ring'' is a [[Set (mathematics)|set]] <math> R </math> equipped with two [[binary operation]]s, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "<math>+</math>" and "<math>\cdot</math>"; e.g. <math>a+b</math> and <math>a \cdot b</math>. 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 [[distributive law|distributes]] over addition; i.e., <math>a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot c\right)</math>. The identity elements for addition and multiplication are denoted <math> 0 </math> and <math> 1 </math>, respectively.

If the multiplication is commutative, i.e.
<math display="block">a \cdot b = b \cdot a,</math>
then the ring <math> R </math> 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 [[integer|ring of integer]]s <math> \mathbb{Z} </math> 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 <math> \mathbb{Z} </math> as an abbreviation of the [[German language|German]] word ''Zahlen'' (numbers).

A [[field (mathematics)|field]] is a commutative ring where <math> 0 \neq 1 </math> and every [[0 (number)|non-zero]] element <math> a </math> is invertible; i.e., has a multiplicative inverse <math> b </math> such that <math> a \cdot b = 1 </math>. Therefore, by definition, any field is a commutative ring. The [[rational number|rational]], [[real number|real]] and [[complex number]]s form fields.

If <math> R </math> is a given commutative ring, then the set of all [[polynomial]]s in the variable <math> X </math> whose coefficients are in <math> R </math> forms the [[polynomial ring]], denoted <math> R \left[ X \right] </math>. The same holds true for several variables.

If <math> V </math> is some [[topological space]], for example a subset of some <math> \mathbb{R}^n </math>, real- or complex-valued [[continuous function]]s on <math> V </math> form a commutative ring. The same is true for [[differentiable function|differentiable]] or [[holomorphic function]]s, when the two concepts are defined, such as for <math> V </math> a [[complex manifold]].

== Divisibility ==
In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of [[divisibility (ring theory)|divisibility for rings]] is richer. An element <math> a </math> of ring <math> R </math> is called a [[unit (algebra)|unit]] if it possesses a multiplicative inverse. Another particular type of element is the [[zero divisor]]s, i.e. an element <math> a </math> such that there exists a non-zero element <math> b </math> of the ring such that <math> ab = 0 </math>. If <math> R </math> possesses no non-zero zero divisors, it is called an [[integral domain]] (or domain). An element <math> a </math> satisfying <math> a^n = 0  </math> for some positive integer <math> n </math> is called [[nilpotent element|nilpotent]].

=== Localizations ===
{{Main|Localization of a ring}}
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 <math> S </math> is a [[multiplicatively closed subset]] of <math> R </math> (i.e. whenever <math> s,t \in S </math> then so is <math> st </math>) then the ''localization'' of <math> R </math> at <math> S </math>, or ''ring of fractions'' with denominators in <math> S </math>, usually denoted <math> S^{-1}R </math> consists of symbols
{{block indent|1= <math>\frac{r}{s}</math> with <math> r \in R, s \in S </math> }}
subject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language <math> \mathbb{Q} </math> is the localization of <math> \mathbb{Z} </math> at all nonzero integers. This construction works for any integral domain <math> R </math> instead of <math> \mathbb{Z} </math>. The localization <math> \left(R\setminus \left\{0\right\}\right)^{-1}R </math> is a field, called the [[quotient field]] of <math> R </math>.

== Ideals and modules ==
{{hatnote|In the following, R denotes a commutative ring.}}

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 ideal|two-sided]], which simplifies the situation considerably.

=== Modules ===
{{Main|Module (mathematics)|l1=Module}}
For a ring <math> R </math>, an <math> R </math>-''module'' <math> M </math> 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 <math> R </math> subject to the same axioms as for a vector space.

The study of modules is significantly more involved than the one of [[vector space]]s, since there are modules that do not have any [[basis (linear algebra)|basis]], that is, do not contain a [[spanning set]] whose elements are [[linearly independent]]s. 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 space]]s in [[linear algebra]]. In particular, [[Noetherian rings]] (see also ''{{slink||Noetherian rings}}'', below) can be defined as the rings such that every submodule of a module of finite type is also of finite type.

=== Ideals ===
{{Main|Ideal (ring theory)|l1=Ideal|Factor ring}}
''Ideals'' of a ring <math> R </math> are the [[submodule]]s of <math> R </math>, i.e., the modules contained in <math> R </math>. In more detail, an ideal <math> I </math> is a non-empty subset of <math> R </math> such that for all <math> r </math> in <math> R </math>, <math> i </math> and <math> j </math> in <math> I </math>, both <math> ri </math> and <math> i+j </math> are in <math> I </math>. 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 [[0 (number)|zero ideal]] <math> \left\{0\right\} </math> and <math> R </math>, the whole ring. These two ideals are the only ones precisely if <math> R </math> is a field. Given any subset <math> F=\left\{f_j\right\}_{j \in J} </math> of <math> R </math> (where <math> J </math> is some index set), the ideal ''generated by'' <math> F </math> is the smallest ideal that contains <math> F </math>. Equivalently, it is given by finite [[linear combination]]s
<math display="block"> r_1 f_1 + r_2 f_2 + \dots + r_n f_n .</math>

==== Principal ideal domains ====
If <math> F </math> consists of a single element <math> r </math>, the ideal generated by <math> F </math> consists of the multiples of <math> r </math>, i.e., the elements of the form <math> rs </math> for arbitrary elements <math> s </math>. Such an ideal is called a [[principal ideal]]. If every ideal is a principal ideal, <math> R </math> is called a [[principal ideal ring]]; two important cases are <math> \mathbb{Z} </math> and <math> k \left[X\right] </math>, the polynomial ring over a field <math> k </math>. These two are in addition domains, so they are called [[principal ideal domain]]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 <math> R </math> 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 <math> a </math> in a domain is called [[irreducible element|irreducible]] if the only way of expressing it as a product
<math display="block"> a=bc ,</math>
is by either <math> b </math> or <math> c </math> being a unit. An example, important in [[Field (mathematics)|field theory]], are [[irreducible polynomial]]s, i.e., irreducible elements in <math> k \left[X\right] </math>, for a field <math> k </math>. The fact that <math> \mathbb{Z} </math> 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 <math> a </math> is a [[prime element]] if whenever <math> a </math> divides a product <math> bc </math>, <math> a </math> divides <math> b </math> or <math> c </math>. In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general.

==== Factor ring ====
The definition of ideals is such that "dividing" <math> I </math> "out" gives another ring, the ''factor ring'' <math> R / I </math>: it is the set of [[coset]]s of <math> I </math> together with the operations
<math display="block"> \left(a+I\right)+\left(b+I\right)=\left(a+b\right)+I </math> and <math> \left(a+I\right) \left(b+I\right)=ab+I </math>.
For example, the ring <math> \mathbb{Z}/n\mathbb{Z} </math> (also denoted <math> \mathbb{Z}_n </math>), where <math> n </math> is an integer, is the ring of integers modulo <math> n </math>. 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 ideal|maximal]]. An ideal <math> m </math> is maximal [[if and only if]] <math> R / m </math> 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 ===
{{Main|Noetherian ring}}
A ring is called ''Noetherian'' (in honor of [[Emmy Noether]], who developed this concept) if every [[ascending chain condition|ascending chain of ideals]]
<math display="block"> 0 \subseteq I_0 \subseteq I_1 \subseteq \dots \subseteq I_n \subseteq I_{n+1} \dots </math>
becomes stationary, i.e. becomes constant beyond some index <math> n </math>. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, [[submodule]]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 <math> R </math> is Noetherian, then so is the polynomial ring <math> R \left[X_1,X_2,\dots,X_n\right] </math> (by [[Hilbert's basis theorem]]), any localization <math> S^{-1}R </math>, and also any factor ring <math> R / I </math>.

Any non-Noetherian ring <math> R </math> is the [[union (set theory)|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 ring|Artinian]] (after [[Emil Artin]]), if every descending chain of ideals
<math display="block"> R \supseteq I_0 \supseteq I_1 \supseteq \dots \supseteq I_n \supseteq I_{n+1} \dots </math>
becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, <math> \mathbb{Z} </math> is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain
<math display="block"> \mathbb{Z} \supsetneq 2\mathbb{Z} \supsetneq 4\mathbb{Z} \supsetneq 8\mathbb{Z} \dots </math>
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.

== Spectrum of a commutative ring ==

=== Prime ideals ===
{{Main|Prime ideal}}
As was mentioned above, <math> \mathbb{Z} </math> is a [[unique factorization domain]]. This is not true for more general rings, as algebraists realized in the 19th century. For example, in
<math display="block">\mathbb{Z}\left[\sqrt{-5}\right]</math>
there are two genuinely distinct ways of writing 6 as a product:
<math display="block">6 = 2 \cdot 3 = \left(1 + \sqrt{-5}\right)\left(1 - \sqrt{-5}\right).</math>
Prime ideals, as opposed to prime elements, provide a way to circumvent this problem. A prime ideal is a proper (i.e., strictly contained in <math> R </math>) ideal <math> p </math> such that, whenever the product <math> ab </math> of any two ring elements <math> a </math> and <math> b </math> is in <math> p, </math> at least one of the two elements is already in <math> p .</math> (The opposite conclusion holds for any ideal, by definition.) Thus, if a prime ideal is principal, it is equivalently generated by a prime element. However, in rings such as <math>\mathbb{Z}\left[\sqrt{-5}\right],</math> prime ideals need not be principal. This limits the usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that in any [[Dedekind ring]] (which includes <math>\mathbb{Z}\left[\sqrt{-5}\right]</math> and more generally the [[algebraic integers|ring of integers in a number field]]) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals.

Any maximal ideal is a prime ideal or, more briefly, is prime. Moreover, an ideal <math>I</math> is prime if and only if the factor ring <math>R/I</math> is an integral domain. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult. Yet another way of expressing the same is to say that the [[Complement (set theory)|complement]] <math>R \setminus p</math> is multiplicatively closed. The localisation <math>\left(R \setminus p\right)^{-1}R</math> is important enough to have its own notation: <math>R_p</math>. This ring has only one maximal ideal, namely <math>pR_p</math>. Such rings are called [[local ring|local]].

=== Spectrum ===
{{Main|Spectrum of a ring}}
[[Image:Spec Z.png|right|400px|thumb|Spec ('''Z''') contains a point for the zero ideal. The closure of this point is the entire space. The remaining points are the ones corresponding to ideals (''p''), where ''p'' is a prime number. These points are closed.]]
The ''spectrum of a ring'' <math>R</math>,{{efn|This notion can be related to the [[Spectrum of an operator|spectrum]] of a linear operator; see ''[[Spectrum of a C*-algebra]]'' and ''[[Gelfand representation]]''.}} denoted by <math>\text{Spec}\ R</math>, is the set of all prime ideals of <math>R</math>. It is equipped with a topology, the [[Zariski topology]], which reflects the algebraic properties of <math>R</math>: a basis of open subsets is given by
<math display="block">D\left(f\right) = \left\{p \in \text{Spec} \ R,f \not\in p\right\},</math> where <math>f</math> is any ring element.
Interpreting <math>f</math> 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 {{nowrap|''R'' → ''R''<sub>''f''</sub>}} and {{nowrap|''R'' → ''R'' / ''fR''}} correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary [[open immersion|open]] and [[closed immersion]]s respectively. Even for basic rings, such as illustrated for {{nowrap|1=''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]] ''k'', mSpec (k[''T''<sub>1</sub>, ..., ''T''<sub>''n''</sub>] / (''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub>)) is in bijection with the set
{{block indent|1= {''x'' =(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) ∊ ''k''<sup>''n''</sup> | ''f''<sub>1</sub>(''x'') = ... = ''f''<sub>''m''</sub>(''x'') = 0.} }}
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]]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 ideal]]s. 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 proceeds by endowing Spec ''R'' with a [[sheaf (mathematics)|sheaf]] <math>\mathcal O</math> (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]]. Given an affine scheme, the underlying ring ''R'' can be recovered as the [[global section]]s of <math>\mathcal O</math>. 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 the opposite direction
{{block indent|1= Spec ''S'' → Spec ''R'', ''q'' ↦ ''f''<sup>−1</sup>(''q''), i.e. any prime ideal of ''S'' is mapped to its [[preimage]] under ''f'', which is a prime ideal of ''R''. }}
The resulting [[equivalence of categories|equivalence]] of the two said categories aptly reflects algebraic properties of rings in a geometrical manner.

Similar to the fact that [[manifold (mathematics)|manifolds]] are locally given by open subsets of '''R'''<sup>''n''</sup>, affine schemes are local models for [[scheme (mathematics)|schemes]], which are the object of study in algebraic geometry. Therefore, several notions concerning commutative rings stem from geometric intuition.

=== Dimension ===
{{Main|Krull 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: {{nowrap|1=dim ''R'' = sup<sub>p ∊ Spec ''R''</sub> dim ''R''<sub>''p''</sub>}}.
* The dimension is independent of nilpotent elements: if {{nowrap|''I'' ⊆ ''R''}} is nilpotent then {{nowrap|1=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 {{nowrap|1=''k''[''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>] = ''n''}}. This axiom is motivated by regarding the polynomial ring in ''n'' variables as an algebraic analogue of [[affine space|''n''-dimensional space]].

The dimension is defined, for any ring ''R'', as the supremum of lengths ''n'' of chains of prime ideals
{{block indent|1= ''p''<sub>0</sub> ⊊ ''p''<sub>1</sub> ⊊ ... ⊊ ''p''<sub>''n''</sub>. }}
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]]. 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]], the [[going-up theorem]] and [[Krull's principal ideal theorem]].

== Ring homomorphisms ==
{{Main|Ring homomorphism}}
A ''ring homomorphism'' or, more colloquially, simply a ''map'', is a map {{nowrap|''f'' : ''R'' → ''S''}} such that
{{block indent|1= ''f''(''a'' + ''b'') = ''f''(''a'') + ''f''(''b''), ''f''(''ab'') = ''f''(''a'')''f''(''b'') and ''f''(1) = 1. }}
These conditions ensure {{nowrap|1=''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
{{block indent|1= ''r'' · ''s'' := ''f''(''r'') · ''s''. }}

The ''kernel'' and ''image'' of ''f'' are defined by {{nowrap|1=ker(''f'') = {{mset|1=''r'' ∈ ''R'', ''f''(''r'') = 0}}}} and {{nowrap|1=im(''f'') = ''f''(''R'') = {{mset|''f''(''r''), ''r'' ∈ ''R''}}}}. The kernel is an [[ring ideal|ideal]] of ''R'', and the image is a [[subring]] 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]], is
<math display="block">\mathbf Z/n = \bigoplus_{i=0}^k \mathbf Z/p_i ,</math>
where {{nowrap|1=''n'' = ''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''k''</sub>}} is a product of pairwise distinct [[prime number]]s.

Commutative rings, together with ring homomorphisms, form a [[category (mathematics)|category]]. The ring '''Z''' is the [[initial object]] in this category, which means that for any commutative ring ''R'', there is a unique ring homomorphism '''Z''' &rarr; ''R''. By means of this map, an integer ''n'' can be regarded as an element of ''R''. For example, the [[binomial formula]]
<math display="block">(a+b)^n = \sum_{k=0}^n \binom n k a^k b^{n-k}</math>
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.

[[File:Tensor product of algebras.png|thumb|The [[universal property]] of {{nowrap|''S'' ⊗<sub>''R''</sub> ''T''}} states that for any two maps {{nowrap|''S'' &rarr; ''W''}} and {{nowrap|''T'' &rarr; ''W''}} which make the outer quadrangle commute, there is a unique map {{nowrap|''S'' ⊗<sub>''R''</sub> ''T'' &rarr; ''W''}} that makes the entire diagram commute.]]
Given two ''R''-algebras ''S'' and ''T'', their [[tensor product of algebras|tensor product]]
{{block indent|1= ''S'' ⊗<sub>''R''</sub> ''T'' }}
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,
{{block indent|1= ''R''[''X''] ⊗<sub>''R''</sub> ''T'' = ''T''[''X'']. }}

=== Finite generation ===
An ''R''-algebra ''S'' is called [[finitely generated algebra|finitely generated (as an algebra)]] if there are finitely many elements ''s''<sub>1</sub>, ..., ''s''<sub>''n''</sub> such that any element of ''s'' is expressible as a polynomial in the ''s''<sub>''i''</sub>. Equivalently, ''S'' is isomorphic to
{{block indent|1= ''R''[''T''<sub>1</sub>, ..., ''T''<sub>''n''</sub>] / ''I''. }}

A much stronger condition is that ''S'' is [[finitely generated module|finitely generated as an ''R''-module]], which means that any ''s'' can be expressed as a ''R''-linear combination of some finite set ''s''<sub>1</sub>, ..., ''s''<sub>''n''</sub>.

== Local rings ==
A ring is called [[local ring|local]] if it has only a single maximal ideal, denoted by ''m''. For any (not necessarily local) ring ''R'', the localization
{{block indent|1= ''R''<sub>''p''</sub> }}
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
{{block indent|1= ''k'' = ''R'' / ''m''. }}
Any ''R''-module ''M'' yields a ''k''-vector space given by {{nowrap|''M'' / ''mM''}}. [[Nakayama's lemma]] shows this passage is preserving important information: a finitely generated module ''M'' is zero if and only if {{nowrap|''M'' / ''mM''}} is zero.

=== Regular local rings ===
[[File:Node_(algebraic_geometry).png|thumb|left|The [[cubic plane curve]] (red) defined by the equation ''y''<sup>2</sup> = ''x''<sup>2</sup>(''x'' + ''1'') is [[singularity (mathematics)|singular]] at the origin, i.e., the ring ''k''[''x'', ''y''] / ''y''<sup>2</sup> &minus; ''x''<sup>2</sup>(''x'' + ''1''), is not a regular ring. The tangent cone (blue) is a union of two lines, which also reflects the singularity.]]
The ''k''-vector space ''m''/''m''<sup>2</sup> is an algebraic incarnation of the [[cotangent space]]. Informally, the elements of ''m'' can be thought of as functions which vanish at the point ''p'', whereas ''m''<sup>2</sup> contains the ones which vanish with order at least 2. For any Noetherian local ring ''R'', the inequality
{{block indent|1= dim<sub>''k''</sub> ''m''/''m''<sup>2</sup> &ge; dim ''R'' }}
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]]. A Noetherian local ring is regular if and only if the ring (which is the ring of functions on the [[tangent cone]])
<math display="block">\bigoplus_n m^n / m^{n+1}</math>
is isomorphic to a polynomial ring over ''k''. Broadly speaking, regular local rings are somewhat similar to polynomial rings.{{sfn|Matsumura|1989|p=143|loc=§7, Remarks|ps=}} Regular local rings are UFD's.{{sfn|Matsumura|1989|loc=§19, Theorem 48|ps=}}

[[Discrete valuation ring]]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]] is a discrete valuation ring.

=== Complete intersections ===
[[File:Twisted_cubic_curve.png|thumb|The [[twisted cubic]] (green) is a set-theoretic complete intersection, but not a complete intersection.]]
By [[Krull's principal ideal theorem]], a foundational result in the [[dimension theory (algebra)|dimension theory of rings]], the dimension of
{{block indent|1= ''R'' = ''k''[''T''<sub>1</sub>, ..., ''T''<sub>''r''</sub>] / (''f''<sub>1</sub>, ..., ''f''<sub>''n''</sub>) }}
is at least ''r'' &minus; ''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.{{sfn|Lyubeznik|1989|ps=}}

=== Cohen–Macaulay rings ===
The [[depth (ring theory)|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''<sub>1</sub>, ..., 
''a''<sub>''n''</sub> &isin; ''m'' such that all ''a''<sub>''i''</sub> are non-zero divisors in
{{block indent|1= ''R'' / (''a''<sub>1</sub>, ..., ''a''<sub>''i''&minus;1</sub>). }}
For any local Noetherian ring, the inequality
{{block indent|1= depth(''R'') &le; dim(''R'') }}
holds. A local ring in which equality takes place is called a [[Cohen–Macaulay ring]]. 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 ring]]s, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.{{sfn|Eisenbud|1995|loc=Corollary 18.10, Proposition 18.13|ps=}}

== 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 [[integral element#Equivalent definitions|integrally closed]] in its [[field of fractions]] is called [[normal ring|normal]]. This is a desirable property, for example any normal one-dimensional ring is necessarily [[Regular local ring|regular]]. Rendering{{clarify|date=March 2012}} a ring normal is known as ''normalization''.

=== Completions ===
If ''I'' is an ideal in a commutative ring ''R'', the powers of ''I'' form [[neighborhood (topology)|topological neighborhoods]] of ''0'' which allow ''R'' to be viewed as a [[topological ring]]. This topology is called the [[I-adic topology|''I''-adic topology]]. ''R'' can then be completed with respect to this topology. Formally, the ''I''-adic completion is the [[inverse limit]] of the rings ''R''/''I<sup>n</sup>''. For example, if ''k'' is a field, ''k''<nowiki>[[</nowiki>''X''<nowiki>]]</nowiki>, the [[formal power series]] 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 number|''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 ring|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]]. {{harvtxt|Hochster|2007}} 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]]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]]s.{{refn|See also [[Serre–Swan theorem]]}} The [[Quillen–Suslin theorem]] asserts that any finitely generated projective module over ''k''[''T''<sub>1</sub>, ..., ''T''<sub>''n''</sub>] (''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]] is finite, say ''n'', which means that any finitely generated ''R''-module has a [[resolution (homological algebra)|resolution]] 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]]. This functor is the [[derived functor]] of the functor
{{block indent|1= Hom<sub>''R''</sub>(''M'', &minus;). }}
The latter functor is exact if ''M'' is projective, but not otherwise: for a surjective map {{nowrap|''E'' &rarr; ''F''}} of ''R''-modules, a map {{nowrap|''M'' &rarr; ''F''}} need not extend to a map {{nowrap|''M'' &rarr; ''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
{{block indent|1= Ext<sup>''n''</sup>(''k'', ''k'') }}
vanishes for all large enough ''n''. Moreover, the dimensions of these Ext-groups, known as [[Betti number]]s, grow polynomially in ''n'' if and only if ''R'' is a [[local complete intersection]] ring.{{sfn|Christensen|Striuli|Veliche|2010|ps=}} 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]] is another non-exact functor relevant in the context of commutative rings: for a general ''R''-module ''M'', the functor
{{block indent|1= ''M'' ⊗<sub>''R''</sub> &minus; }}
is only right exact. If it is exact, ''M'' is called [[flat module|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
{{block indent|1= ''S'' / ''pS'' = ''S'' ⊗<sub>''R''</sub> ''R'' / ''p'' }}
(for prime ideals ''p'' in ''R'') have the "expected" dimension, namely {{nowrap|dim ''S'' &minus; dim ''R'' + dim(''R'' / ''p'')}}.

== Properties ==
By [[Wedderburn's little theorem|Wedderburn's theorem]], every finite [[division ring]] is commutative, and therefore a [[finite field]]. Another condition ensuring commutativity of a ring, due to [[Nathan Jacobson|Jacobson]], is the following: for every element ''r'' of ''R'' there exists an integer {{nowrap|''n'' > 1}} such that {{nowrap|1=''r''<sup>''n''</sup> = ''r''}}.{{sfn|Jacobson|1945|ps=}} If, {{nowrap|1=''r''<sup>2</sup> = ''r''}} for every ''r'', the ring is called [[Boolean ring]]. More general conditions which guarantee commutativity of a ring are also known.{{sfn|Pinter-Lucke|2007|ps=}}

== Generalizations ==

=== Graded-commutative rings ===
[[File:Pair_of_pants.png|thumb|A [[pair of pants (mathematics)|pair of pants]] is a [[cobordism]] between a circle and two disjoint circles. Cobordism classes, with the [[cartesian product]] as multiplication and [[disjoint union]] as the sum, form the [[cobordism ring]].]]
A [[graded ring]] {{nowrap|1=''R'' = ⨁<sub>''i''∊'''Z'''</sub> ''R''<sub>''i''</sub>}} is called [[graded-commutative ring|graded-commutative]] if, for all homogeneous elements ''a'' and ''b'',
{{block indent|1= ''ab'' = (&minus;1)<sup>deg ''a'' ⋅ deg ''b''</sup> ''ba''. }}
If the ''R''<sub>''i''</sub> are connected by differentials ∂ such that an abstract form of the [[product rule]] holds, i.e.,
{{block indent|1= ∂(''ab'') = ∂(''a'')''b'' + (&minus;1)<sup>deg ''a''</sup>a∂(''b''), }}
''R'' is called a [[differential graded algebra|commutative differential graded algebra]] (cdga). An example is the complex of [[differential form]]s on a [[manifold (mathematics)|manifold]], with the multiplication given by the [[exterior product]], 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's universal ring|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]].

A related notion is an [[almost commutative ring]], which means that ''R'' is [[filtration (mathematics)|filtered]] in such a way that the associated graded ring
{{block indent|1= gr ''R'' := ⨁ ''F''<sub>''i''</sub>''R'' / ⨁ ''F''<sub>''i''&minus;1</sub>''R'' }}
is commutative. An example is the [[Weyl algebra]] and more general rings of [[differential operator]]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-infinity ring|E<sub>∞</sub>-ring]].

== Applications of the commutative rings ==
* [[Holomorphic function]]s
* [[Algebraic K-theory]]
* [[Topological K-theory]]
* [[Divided power structure]]s
* [[Witt vector]]s
* [[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]]

== See also ==
* [[Almost ring]], a certain generalization of a commutative ring
* [[Divisibility (ring theory)]]: [[nilpotent element]], (ex. [[dual number]]s)
* Ideals and modules: [[Radical of an ideal]], [[Morita equivalence]]
* [[Ring homomorphism]]s: [[integral element]]: [[Cayley–Hamilton theorem]], [[Integrally closed domain]], [[Krull ring]], [[Krull–Akizuki theorem]], [[Mori–Nagata theorem]]
* Primes: [[Prime avoidance lemma]], [[Jacobson radical]], [[Nilradical of a ring]], Spectrum: [[Compact space]], [[Connected ring]], [[Differential calculus over commutative algebras]], [[Banach–Stone theorem]]
* [[Local ring]]s: [[Gorenstein local ring]] (also used in [[Wiles's proof of Fermat's Last Theorem]]): [[Duality (mathematics)]], [[Eben Matlis]]; [[Dualizing module]], [[Popescu's theorem]], [[Artin approximation theorem]].

== Notes ==
{{notelist}}

== Citations ==
{{reflist}}

== References ==
{{refbegin}}
* {{Citation | last1=Christensen | first1=Lars Winther | first2=Janet | last2=Striuli | first3=Oana | last3=Veliche | title=Growth in the minimal injective resolution of a local ring | journal=Journal of the London Mathematical Society | series=Second Series | volume=81 | issue=1 | pages=24–44 | year=2010 | doi=10.1112/jlms/jdp058 | arxiv=0812.4672| s2cid=14764965 }}
* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra. With a view toward algebraic geometry. | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150}}
* {{Citation | last=Hochster | first=Melvin | title=Homological conjectures, old and new | journal=Illinois J. Math. | volume=51 | issue=1 | year=2007 | pages=151–169 | doi=10.1215/ijm/1258735330| doi-access=free }}
* {{Citation | doi=10.2307/1969205 | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=Structure theory of algebraic algebras of bounded degree | year=1945 | journal=[[Annals of Mathematics]] | issn=0003-486X | volume=46 | issue=4 | pages=695–707 | jstor=1969205}}
* {{Citation | last1=Lyubeznik |first1=Gennady | chapter=A survey of problems and results on the number of defining equations | title=Representations, resolutions and intertwining numbers | pages=375–390 | year=1989 | zbl=0753.14001}}
* {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Ring Theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | year=1989}}
* {{Citation | last1=Pinter-Lucke | first1=James | title=Commutativity conditions for rings: 1950–2005 | doi=10.1016/j.exmath.2006.07.001 | year=2007 | journal=Expositiones Mathematicae | issn=0723-0869 | volume=25 | issue=2 | pages=165–174| doi-access=free }}
{{refend}}

== Further reading ==
{{refbegin}}
* {{Citation | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | last2=Macdonald | first2=I. G. | author2-link=Ian G. Macdonald | title=Introduction to commutative algebra | publisher=Addison-Wesley Publishing Co. | year=1969 }}
* {{Citation | last1=Balcerzyk | first1=Stanisław | last2=Józefiak | first2=Tadeusz | title=Commutative Noetherian and Krull rings | publisher=Ellis Horwood Ltd. | location=Chichester | series=Ellis Horwood Series: Mathematics and its Applications | isbn=978-0-13-155615-7 | year=1989}}
* {{Citation | last1=Balcerzyk | first1=Stanisław | last2=Józefiak | first2=Tadeusz | title=Dimension, multiplicity and homological methods | publisher=Ellis Horwood Ltd. | location=Chichester | series=Ellis Horwood Series: Mathematics and its Applications.  | isbn=978-0-13-155623-2 | year=1989}}
* {{Citation | last1=Kaplansky | first1=Irving | author1-link=Irving Kaplansky | title=Commutative rings | publisher=[[University of Chicago Press]] | edition=Revised | mr=0345945 | year=1974}}
* {{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=Local rings | publisher=Interscience Publishers  | series=Interscience Tracts in Pure and Applied Mathematics | isbn=978-0-88275-228-0 |year=1975 | mr=0155856 | orig-year=1962 | volume=13 | pages=xiii+234}}
* {{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative Algebra I, II | publisher= D. van Nostrand, Inc. | location=Princeton, N.J. | series=University series in Higher Mathematics | year=1958–60}} ''(Reprinted 1975–76 by Springer as volumes 28–29 of Graduate Texts in Mathematics.)''
{{refend}}

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[[Category:Commutative algebra]]
[[Category:Ring theory]]
[[Category:Algebraic structures]]