commutative ring

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In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative.

# Definition and first examples

## Definition

A ''ring'' is a $R$ equipped with two 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 under addition as well as a under multiplication, where multiplication 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 s $\mathbb$ 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 word ''Zahlen'' (numbers). A is a commutative ring where $0 \not = 1$ and every 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 , and s form fields. If ''$R$'' is a given commutative ring, then the set of all s in the variable $X$ whose coefficients are in ''$R$'' forms the , denoted $R \left\left[ X \right\right]$. The same holds true for several variables. If ''$V$'' is some , for example a subset of some $\mathbb^n$, real- or complex-valued s on ''$V$'' form a commutative ring. The same is true for or s, when the two concepts are defined, such as for ''$V$'' a .

# Divisibility

In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of is richer. An element $a$ of ring ''$R$'' is called a if it possesses a multiplicative inverse. Another particular type of element is the 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 (or domain). An element $a$ satisfying $a^n = 0$ for some positive integer $n$ is called .

## 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 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 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 , which simplifies the situation considerably.

## Modules and ideals

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 spaces in , since several features of vector spaces fail for modules in general: modules need not be , i.e., of the form $M= \bigoplus_ R.$ Even for free modules, the (i.e. the analog of the dimension of vector spaces) may not be well-defined. Finally, submodules of finitely generated modules need not be finitely generated (unless ''$R$'' is Noetherian, see ).

## Ideals

''Ideals'' of a ring ''$R$'' are the 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 ''$\left\$'' and ''$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 s ''$r_1 f_1 + r_2 f_2 + \dots + r_n f_n .$''

### 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 . If every ideal is a principal ideal, ''$R$'' is called a ; two important cases are ''$\mathbb$'' and ''$k \left\left[X\right\right]$'', the polynomial ring over a field ''$k$''. These two are in addition domains, so they are called 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 (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 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 , are s, i.e., irreducible elements in ''$k \left\left[X\right\right]$'', 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 . An element ''$a$'' is a 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 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 . 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 . An ideal ''$m$'' is maximal ''$R$'' / ''$m$'' is a field. Except for the , any ring (with identity) possesses at least one maximal ideal; this follows from .

## Noetherian rings

A ring is called ''Noetherian'' (in honor of , who developed this concept) if every ''$0 \subseteq I_0 \subseteq I_1 \subseteq \dots \subseteq I_n \subseteq I_ \dots$'' becomes stationary, i.e. becomes constant beyond some index ''$n$''. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, 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 ''$R \left\left[X_1,X_2,\dots,X_n\right\right]$'' (by ), any localization ''$S^R$'', and also any factor ring ''$R$'' / ''$I$''. Any non-Noetherian ring ''$R$'' is the of its Noetherian subrings. This fact, known as , allows the extension of certain theorems to non-Noetherian rings.

## Artinian rings

A ring is called (after ), 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 , 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 . This is not true for more general rings, as algebraists realized in the 19th century. For example, in $\mathbb\left[\sqrt\right]$ there are two genuinely distinct ways of writing 6 as a product: $6 = 2 \cdot 3 = \left(1 + \sqrt\right)\left(1 - \sqrt\right).$ 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 ''$R$'') ideal ''$p$'' such that, whenever the product ''$ab$'' of any two ring elements ''$a$'' and ''$b$'' is in ''$p$'', at least one of the two elements is already in ''$p$''. (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 $\mathbb\left\left[\sqrt\right\right]$, 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 (which includes $\mathbb\left\left[\sqrt\right\right]$ and more generally the ) 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 ''$I$'' is prime if and only if the factor ring ''$R$'' / ''$I$'' 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 ''$R \backslash p$'' is multiplicatively closed. The localisation ''$\left\left(R \backslash p\right\right)^R$''(''R'' \ ''p'')−1''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 .

## The spectrum

The ''spectrum of a ring $R$'',This notion can be related to the of a linear operator, see and . denoted by ''$\text\ R$'', is the set of all prime ideals of ''$R$''. It is equipped with a topology, the , 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 and s 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 ''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 s of Spec ''R''. For a Noetherian ring ''R'', Spec ''R'' has only finitely many irreducible components. This is a geometric restatement of , according to which any ideal can be decomposed as a product of finitely many 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 proceeds by endowing Spec ''R'' with a $\mathcal O$ (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an . Given an affine scheme, the underlying ring ''R'' can be recovered as the s 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 in the opposite direction The resulting of the two said categories aptly reflects algebraic properties of rings in a geometrical manner. Similar to the fact that are locally given by open subsets of R''n'', affine schemes are local models for , 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 . 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 . 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 , the and .

# 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 of ''R'', and the image is a of ''S''. A ring homomorphism is called an isomorphism if it is bijective. An example of a ring isomorphism, known as 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 s. Commutative rings, together with ring homomorphisms, form a . The ring Z is the 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 $(a+b)^n = \sum_^n \binom n k a^k b^$ 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 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 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 , 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 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 of ''R'' is defined as Any ''R''-module ''M'' yields a ''k''-vector space given by ''M'' / ''mM''. 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 . 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 . A Noetherian local ring is regular if and only if the ring (which is the ring of functions on the ) $\bigoplus_n m^n / m^$ 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. 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 is a discrete valuation ring.

## Complete intersections

By , a foundational result in the , the dimension of is at least ''r'' − ''n''. A ring ''R'' is called a 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 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 . 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 s, 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 in its is called . This is a desirable property, for example any normal one-dimensional ring is necessarily . Rendering a ring normal is known as ''normalization''.

## Completions

If ''I'' is an ideal in a commutative ring ''R'', the powers of ''I'' form of ''0'' which allow ''R'' to be viewed as a . This topology is called the . ''R'' can then be completed with respect to this topology. Formally, the ''I''-adic completion is the of the rings ''R''/''In''. For example, if ''k'' is a field, ''k'', the 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 is the completion of Z with respect to the principal ideal (''p''). Any ring that is isomorphic to its own completion, is called . Complete local rings satisfy , 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 . lists some open questions in this area of active research.

## Projective modules and Ext functors

Projective modules can be defined to be the 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 s. The 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 is finite, say ''n'', which means that any finitely generated ''R''-module has a 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 . This functor is the 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 s, grow polynomially in ''n'' if and only if ''R'' is a ring. A key argument in such considerations is the , which provides an explicit free resolution of the residue field ''k'' of a local ring ''R'' in terms of a regular sequence.

## Flatness

The 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 . 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 , every finite is commutative, and therefore a . Another condition ensuring commutativity of a ring, due to , 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 . More general conditions which guarantee commutativity of a ring are also known.

# Generalizations

A ''R'' = ⨁''i''∊Z ''R''''i'' is called if If the ''R''''i'' are connected by differentials ∂ such that an abstract form of the holds, i.e., ''R'' is called a (cdga). An example is the complex of s on a , with the multiplication given by the , is a cdga. The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the . A broad range examples of graded rings arises in this way. For example, the 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 . A related notion is an , which means that ''R'' is in such a way that the associated graded ring is commutative. An example is the and more general rings of s.

## Simplicial commutative rings

A is a in the category of commutative rings. They are building blocks for (connective) . A closely related but more general notion is that of .

* , a certain generalization of a commutative ring. * : , example: s * Ideals and modules: , * Ring homomorphisms: : , , , * Primes: , , , Spectrum: , , , * Local rings: : , ; , , . * "Applications" (commutative rings arising in mathematics): s, , , s, s, , , , (of a commutative group), see also

* * * * * * *