Ring (mathematics)

Equip the set <math>\Z /4\Z = \left\{\overline{0}, \overline{1}, \overline{2}, \overline{3}\right\}</math> with the following operations:

The sum <math>\overline{x} + \overline{y}</math> in Template:Tmath is the remainder when the integer Template:Math is divided by Template:Math (as Template:Math is always smaller than Template:Math, this remainder is either Template:Math or Template:Math). For example, <math>\overline{2} + \overline{3} = \overline{1}</math> and <math>\overline{3} + \overline{3} = \overline{2}.</math>
The product <math>\overline{x} \cdot \overline{y}</math> in Template:Tmath is the remainder when the integer Template:Mvar is divided by Template:Math. For example, <math>\overline{2} \cdot \overline{3} = \overline{2}</math> and <math>\overline{3} \cdot \overline{3} = \overline{1}.</math>

Then Template:Tmath is a ring: each axiom follows from the corresponding axiom for Template:Tmath If Template:Mvar is an integer, the remainder of Template:Mvar when divided by Template:Math may be considered as an element of Template:Tmath and this element is often denoted by "Template:Math" or <math>\overline x,</math> which is consistent with the notation for Template:Math. The additive inverse of any <math>\overline x</math> in Template:Tmath is <math>-\overline x=\overline{-x}.</math> For example, <math>-\overline{3} = \overline{-3} = \overline{1}.</math>

Template:Tmath has a subring Template:Tmath, and if <math>p</math> is prime, then Template:Tmath has no subrings.

Example: 2-by-2 matricesEdit

The set of 2-by-2 square matrices with entries in a field Template:Mvar isTemplate:Sfnp Template:Sfnp Template:Sfnp Template:Sfnp

<math>\operatorname{M}_2(F) = \left\{ \left.\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right|\ a, b, c, d \in F \right\}.</math>

With the operations of matrix addition and matrix multiplication, <math>\operatorname{M}_2(F)</math> satisfies the above ring axioms. The element <math>\left( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)</math> is the multiplicative identity of the ring. If <math>A = \left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right)</math> and <math>B = \left( \begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix} \right),</math> then <math>AB = \left( \begin{smallmatrix} 0 & 0 \\ 0 & 1 \end{smallmatrix} \right)</math> while <math>BA = \left( \begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix} \right);</math> this example shows that the ring is noncommutative.

More generally, for any ring Template:Mvar, commutative or not, and any nonnegative integer Template:Mvar, the square Template:Math matrices with entries in Template:Mvar form a ring; see Matrix ring.

HistoryEdit

File:Dedekind.jpeg

Richard Dedekind, one of the founders of ring theory

DedekindEdit

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers.<ref name="history">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In 1871, Richard Dedekind defined the concept of the ring of integers of a number field.Template:Sfnp In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.

HilbertEdit

The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897.Template:Sfnp In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring),Template:CN so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence).Template:Sfnp Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if Template:Math then:

<math>\begin{align}

a^3 &= 4a-1, \\ a^4 &= 4a^2-a, \\ a^5 &= -a^2+16a-4, \\ a^6 &= 16a^2-8a+1, \\ a^7 &= -8a^2+65a-16, \\ \vdots \ & \qquad \vdots \end{align}</math> and so on; in general, Template:Math is going to be an integral linear combination of Template:Math, Template:Math, and Template:Math.

Fraenkel and NoetherEdit

The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915,Template:Sfnp Template:Sfnp but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse.Template:Sfnp In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.Template:Sfnp

Multiplicative identity and the term "ring"Edit

Fraenkel applied the term "ring" to structures with axioms that included a multiplicative identity,Template:Sfnp whereas Noether applied it to structures that did not.Template:Sfnp

Most or all books on algebraTemplate:Sfnp Template:Sfnp up to around 1960 followed Noether's convention of not requiring a Template:Math for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of Template:Math in the definition of "ring", especially in advanced books by notable authors such as Artin,Template:Sfnp Bourbaki,Template:Sfnp Eisenbud,Template:Sfnp and Lang.Template:Sfnp There are also books published as late as 2022 that use the term without the requirement for a Template:Math.Template:Sfnp Template:Sfnp Template:Sfnp Template:Sfnp Likewise, the Encyclopedia of Mathematics does not require unit elements in rings.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In a research article, the authors often specify which definition of ring they use in the beginning of that article.

Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a Template:Math, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."Template:Sfnp Poonen makes the counterargument that the natural notion for rings would be the direct product rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.Template:Efn Template:Sfnp

Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention:

to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit",Template:Sfnp or "ring with 1".Template:Sfnp
to omit a requirement for a multiplicative identity: "rng"Template:Sfnp or "pseudo-ring",Template:Sfnp although the latter may be confusing because it also has other meanings.

Basic examplesEdit

Commutative ringsEdit

The prototypical example is the ring of integers with the two operations of addition and multiplication.
The rational, real and complex numbers are commutative rings of a type called fields.
A unital associative algebra over a commutative ring Template:Mvar is itself a ring as well as an [[module (mathematics)|Template:Mvar-module]]. Some examples:
- The algebra Template:Math of polynomials with coefficients in Template:Mvar.
- The algebra <math>RX_1, \dots, X_n</math> of formal power series with coefficients in Template:Mvar.
- The set of all continuous real-valued functions defined on the real line forms a commutative Template:Tmath-algebra. The operations are pointwise addition and multiplication of functions.
- Let Template:Mvar be a set, and let Template:Mvar be a ring. Then the set of all functions from Template:Mvar to Template:Mvar forms a ring, which is commutative if Template:Mvar is commutative.
The ring of quadratic integers, the integral closure of Template:Tmath in a quadratic extension of Template:Tmath It is a subring of the ring of all algebraic integers.
The ring of profinite integers Template:Tmath the (infinite) product of the rings of Template:Mvar-adic integers Template:Tmath over all prime numbers Template:Mvar.
The Hecke ring, the ring generated by Hecke operators.
If Template:Mvar is a set, then the power set of Template:Mvar becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.

Noncommutative ringsEdit

For any ring Template:Mvar and any natural number Template:Mvar, the set of all square Template:Mvar-by-Template:Mvar matrices with entries from Template:Mvar, forms a ring with matrix addition and matrix multiplication as operations. For Template:Math, this matrix ring is isomorphic to Template:Mvar itself. For Template:Math (and Template:Mvar not the zero ring), this matrix ring is noncommutative.
If Template:Math is an abelian group, then the endomorphisms of Template:Math form a ring, the endomorphism ring Template:Math of Template:Math. The operations in this ring are addition and composition of endomorphisms. More generally, if Template:Mvar is a left module over a ring Template:Mvar, then the set of all Template:Mvar-linear maps forms a ring, also called the endomorphism ring and denoted by Template:Math.
The endomorphism ring of an elliptic curve. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
If Template:Math is a group and Template:Mvar is a ring, the group ring of Template:Math over Template:Mvar is a free module over Template:Mvar having Template:Math as basis. Multiplication is defined by the rules that the elements of Template:Math commute with the elements of Template:Mvar and multiply together as they do in the group Template:Math.
The ring of differential operators (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most Banach algebras are noncommutative.

Non-ringsEdit

The set of natural numbers Template:Tmath with the usual operations is not a ring, since Template:Tmath is not even a group (not all the elements are invertible with respect to addition – for instance, there is no natural number which can be added to Template:Math to get Template:Math as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers Template:Tmath The natural numbers (including Template:Math) form an algebraic structure known as a semiring (which has all of the axioms of a ring excluding that of an additive inverse).
Let Template:Mvar be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as convolution: <math display="block">(f * g)(x) = \int_{-\infty}^\infty f(y)g(x - y) \, dy.</math> Then Template:Mvar is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of Template:Mvar.

Basic conceptsEdit

Products and powersEdit

For each nonnegative integer Template:Mvar, given a sequence Template:Tmath of Template:Mvar elements of Template:Mvar, one can define the product Template:Tmath recursively: let Template:Math and let Template:Math for Template:Math.

As a special case, one can define nonnegative integer powers of an element Template:Mvar of a ring: Template:Math and Template:Math for Template:Math. Then Template:Math for all Template:Math.

Elements in a ringEdit

A left zero divisor of a ring Template:Mvar is an element Template:Mvar in the ring such that there exists a nonzero element Template:Mvar of Template:Mvar such that Template:Math.Template:Efn A right zero divisor is defined similarly.

A nilpotent element is an element Template:Mvar such that Template:Math for some Template:Math. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.

An idempotent <math>e</math> is an element such that Template:Math. One example of an idempotent element is a projection in linear algebra.

A unit is an element Template:Mvar having a multiplicative inverse; in this case the inverse is unique, and is denoted by Template:Math. The set of units of a ring is a group under ring multiplication; this group is denoted by Template:Math or Template:Math or Template:Math. For example, if Template:Mvar is the ring of all square matrices of size Template:Mvar over a field, then Template:Math consists of the set of all invertible matrices of size Template:Mvar, and is called the general linear group.

SubringEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A subset Template:Mvar of Template:Mvar is called a subring if any one of the following equivalent conditions holds:

the addition and multiplication of Template:Mvar restrict to give operations Template:Math making Template:Mvar a ring with the same multiplicative identity as Template:Mvar.
Template:Math; and for all Template:Mvar in Template:Mvar, the elements Template:Mvar, Template:Math, and Template:Mvar are in Template:Mvar.
Template:Mvar can be equipped with operations making it a ring such that the inclusion map Template:Math is a ring homomorphism.

For example, the ring Template:Tmath of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Template:Tmath (in both cases, Template:Tmath contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers Template:Tmath does not contain the identity element Template:Math and thus does not qualify as a subring of Template:Tmath one could call Template:Tmath a subrng, however.

An intersection of subrings is a subring. Given a subset Template:Mvar of Template:Mvar, the smallest subring of Template:Mvar containing Template:Mvar is the intersection of all subrings of Template:Mvar containing Template:Mvar, and it is called the subring generated by Template:Math.

For a ring Template:Mvar, the smallest subring of Template:Mvar is called the characteristic subring of Template:Mvar. It can be generated through addition of copies of Template:Math and Template:Math. It is possible that Template:Math (Template:Mvar times) can be zero. If Template:Mvar is the smallest positive integer such that this occurs, then Template:Mvar is called the characteristic of Template:Mvar. In some rings, Template:Math is never zero for any positive integer Template:Mvar, and those rings are said to have characteristic zero.

Given a ring Template:Mvar, let Template:Math denote the set of all elements Template:Mvar in Template:Mvar such that Template:Mvar commutes with every element in Template:Mvar: Template:Math for any Template:Mvar in Template:Mvar. Then Template:Math is a subring of Template:Mvar, called the center of Template:Mvar. More generally, given a subset Template:Mvar of Template:Mvar, let Template:Mvar be the set of all elements in Template:Mvar that commute with every element in Template:Mvar. Then Template:Mvar is a subring of Template:Mvar, called the centralizer (or commutant) of Template:Mvar. The center is the centralizer of the entire ring Template:Mvar. Elements or subsets of the center are said to be central in Template:Mvar; they (each individually) generate a subring of the center.

IdealEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Let Template:Mvar be a ring. A left ideal of Template:Mvar is a nonempty subset Template:Mvar of Template:Mvar such that for any Template:Mvar in Template:Mvar and Template:Mvar in Template:Mvar, the elements Template:Math and Template:Mvar are in Template:Mvar. If Template:Mvar denotes the Template:Mvar-span of Template:Mvar, that is, the set of finite sums

<math>r_1 x_1 + \cdots + r_n x_n \quad \textrm{such}\;\textrm{that}\; r_i \in R \; \textrm{ and } \; x_i \in I,</math>

then Template:Mvar is a left ideal if Template:Math. Similarly, a right ideal is a subset Template:Mvar such that Template:Math. A subset Template:Mvar is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of Template:Mvar. If Template:Mvar is a subset of Template:Mvar, then Template:Math is a left ideal, called the left ideal generated by Template:Mvar; it is the smallest left ideal containing Template:Mvar. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of Template:Mvar.

If Template:Mvar is in Template:Mvar, then Template:Math and Template:Math are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by Template:Mvar. The principal ideal Template:Math is written as Template:Math. For example, the set of all positive and negative multiples of Template:Math along with Template:Math form an ideal of the integers, and this ideal is generated by the integer Template:Math. In fact, every ideal of the ring of integers is principal.

Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.

Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian.

For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal Template:Mvar of Template:Mvar is called a prime ideal if for any elements <math>x, y\in R</math> we have that <math>xy \in P</math> implies either <math>x \in P</math> or <math>y\in P.</math> Equivalently, Template:Mvar is prime if for any ideals Template:Math, Template:Math we have that Template:Math implies either Template:Math or Template:Math. This latter formulation illustrates the idea of ideals as generalizations of elements.

HomomorphismEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A homomorphism from a ring Template:Math to a ring Template:Math is a function Template:Mvar from Template:Mvar to Template:Mvar that preserves the ring operations; namely, such that, for all Template:Math, Template:Math in Template:Mvar the following identities hold:

<math>\begin{align}

& f(a+b) = f(a) \ddagger f(b) \\ & f(a\cdot b) = f(a)*f(b) \\ & f(1_R) = 1_S \end{align}</math>

If one is working with Template:Nat, then the third condition is dropped.

A ring homomorphism Template:Mvar is said to be an isomorphism if there exists an inverse homomorphism to Template:Mvar (that is, a ring homomorphism that is an inverse function), or equivalently if it is bijective.

Examples:

The function that maps each integer Template:Mvar to its remainder modulo Template:Math (a number in Template:Math) is a homomorphism from the ring Template:Tmath to the quotient ring Template:Tmath ("quotient ring" is defined below).
If Template:Mvar is a unit element in a ring Template:Mvar, then <math>R \to R, x \mapsto uxu^{-1}</math> is a ring homomorphism, called an inner automorphism of Template:Mvar.
Let Template:Mvar be a commutative ring of prime characteristic Template:Mvar. Then Template:Math is a ring endomorphism of Template:Mvar called the Frobenius homomorphism.
The Galois group of a field extension Template:Math is the set of all automorphisms of Template:Mvar whose restrictions to Template:Mvar are the identity.
For any ring Template:Mvar, there are a unique ring homomorphism Template:Tmath and a unique ring homomorphism Template:Math.
An epimorphism (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map Template:Tmath is an epimorphism.
An algebra homomorphism from a Template:Mvar-algebra to the endomorphism algebra of a vector space over Template:Mvar is called a representation of the algebra.

Given a ring homomorphism Template:Math, the set of all elements mapped to 0 by Template:Mvar is called the kernel of Template:Mvar. The kernel is a two-sided ideal of Template:Mvar. The image of Template:Mvar, on the other hand, is not always an ideal, but it is always a subring of Template:Mvar.

To give a ring homomorphism from a commutative ring Template:Mvar to a ring Template:Mvar with image contained in the center of Template:Mvar is the same as to give a structure of an algebra over Template:Mvar to Template:Mvar (which in particular gives a structure of an Template:Mvar-module).

Quotient ringEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The notion of quotient ring is analogous to the notion of a quotient group. Given a ring Template:Math and a two-sided ideal Template:Mvar of Template:Math, view Template:Mvar as subgroup of Template:Math; then the quotient ring Template:Math is the set of cosets of Template:Mvar together with the operations

<math>\begin{align}

& (a+I)+(b+I) = (a+b)+I, \\ & (a+I)(b+I) = (ab)+I. \end{align}</math> for all Template:Math in Template:Mvar. The ring Template:Math is also called a factor ring.

As with a quotient group, there is a canonical homomorphism Template:Math, given by Template:Math. It is surjective and satisfies the following universal property:

If Template:Math is a ring homomorphism such that Template:Math, then there is a unique homomorphism <math>\overline{f} : R/I \to S</math> such that <math>f = \overline{f} \circ p.</math>

For any ring homomorphism Template:Math, invoking the universal property with Template:Math produces a homomorphism <math>\overline{f} : R / \ker f \to S</math> that gives an isomorphism from Template:Math to the image of Template:Mvar.

ModulesEdit

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The concept of a module over a ring generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring Template:Mvar, an Template:Mvar-module Template:Mvar is an abelian group equipped with an operation Template:Math (associating an element of Template:Mvar to every pair of an element of Template:Mvar and an element of Template:Mvar) that satisfies certain axioms. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all Template:Math, Template:Math in Template:Mvar and all Template:Math, Template:Math in Template:Mvar,

Template:Mvar is an abelian group under addition.

<math>\begin{align}

& a(x+y) = ax+ay \\ & (a+b)x = ax+bx \\ & 1x = x \\ & (ab)x = a(bx) \end{align}</math> When the ring is noncommutative these axioms define left modules; right modules are defined similarly by writing Template:Mvar instead of Template:Mvar. This is not only a change of notation, as the last axiom of right modules (that is Template:Math) becomes Template:Math, if left multiplication (by ring elements) is used for a right module.

Basic examples of modules are ideals, including the ring itself.

Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the dimension of a vector space). In particular, not all modules have a basis.

The axioms of modules imply that Template:Math, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.

Any ring homomorphism induces a structure of a module: if Template:Math is a ring homomorphism, then Template:Mvar is a left module over Template:Mvar by the multiplication: Template:Math. If Template:Mvar is commutative or if Template:Math is contained in the center of Template:Mvar, the ring Template:Mvar is called a Template:Mvar-algebra. In particular, every ring is an algebra over the integers.

ConstructionsEdit

Direct productEdit

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Let Template:Mvar and Template:Mvar be rings. Then the product Template:Math can be equipped with the following natural ring structure:

<math>\begin{align}

& (r_1,s_1) + (r_2,s_2) = (r_1+r_2,s_1+s_2) \\ & (r_1,s_1) \cdot (r_2,s_2)=(r_1\cdot r_2,s_1\cdot s_2) \end{align}</math> for all Template:Math in Template:Mvar and Template:Math in Template:Mvar. The ring Template:Math with the above operations of addition and multiplication and the multiplicative identity Template:Math is called the direct product of Template:Mvar with Template:Mvar. The same construction also works for an arbitrary family of rings: if Template:Mvar are rings indexed by a set Template:Mvar, then <math display="inline"> \prod_{i \in I} R_i</math> is a ring with componentwise addition and multiplication.

Let Template:Mvar be a commutative ring and <math>\mathfrak{a}_1, \cdots, \mathfrak{a}_n</math> be ideals such that <math>\mathfrak{a}_i + \mathfrak{a}_j = (1)</math> whenever Template:Math. Then the Chinese remainder theorem says there is a canonical ring isomorphism: <math display="block">R /{\textstyle \bigcap_{i=1}^{n}{\mathfrak{a}_i}} \simeq \prod_{i=1}^{n}{R/ \mathfrak{a}_i}, \qquad x \bmod {\textstyle \bigcap_{i=1}^{n}\mathfrak{a}_i} \mapsto (x \bmod \mathfrak{a}_1, \ldots , x \bmod \mathfrak{a}_n).</math>

A "finite" direct product may also be viewed as a direct sum of ideals.Template:Sfnp Namely, let <math>R_i, 1 \le i \le n</math> be rings, <math display="inline">R_i \to R = \prod R_i</math> the inclusions with the images <math>\mathfrak{a}_i</math> (in particular <math>\mathfrak{a}_i</math> are rings though not subrings). Then <math>\mathfrak{a}_i</math> are ideals of Template:Mvar and <math display="block">R = \mathfrak{a}_1 \oplus \cdots \oplus \mathfrak{a}_n, \quad \mathfrak{a}_i \mathfrak{a}_j = 0, i \ne j, \quad \mathfrak{a}_i^2 \subseteq \mathfrak{a}_i</math> as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to Template:Mvar. Equivalently, the above can be done through central idempotents. Assume that Template:Mvar has the above decomposition. Then we can write <math display="block">1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak{a}_i.</math> By the conditions on <math>\mathfrak{a}_i,</math> one has that Template:Mvar are central idempotents and Template:Math, Template:Math (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let <math>\mathfrak{a}_i = R e_i,</math> which are two-sided ideals. If each Template:Mvar is not a sum of orthogonal central idempotents,Template:Efn then their direct sum is isomorphic to Template:Mvar.

An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).

Polynomial ringEdit

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Given a symbol Template:Mvar (called a variable) and a commutative ring Template:Mvar, the set of polynomials

<math>R[t] = \left\{ a_n t^n + a_{n-1} t^{n -1} + \dots + a_1 t + a_0 \mid n \ge 0, a_j \in R \right\}</math>

forms a commutative ring with the usual addition and multiplication, containing Template:Mvar as a subring. It is called the polynomial ring over Template:Mvar. More generally, the set <math>R\left[t_1, \ldots, t_n\right]</math> of all polynomials in variables <math>t_1, \ldots, t_n</math> forms a commutative ring, containing <math>R\left[t_i\right]</math> as subrings.

If Template:Mvar is an integral domain, then Template:Math is also an integral domain; its field of fractions is the field of rational functions. If Template:Mvar is a Noetherian ring, then Template:Math is a Noetherian ring. If Template:Mvar is a unique factorization domain, then Template:Math is a unique factorization domain. Finally, Template:Mvar is a field if and only if Template:Math is a principal ideal domain.

Let <math>R \subseteq S</math> be commutative rings. Given an element Template:Mvar of Template:Mvar, one can consider the ring homomorphism

<math>R[t] \to S, \quad f \mapsto f(x)</math>

(that is, the substitution). If Template:Math and Template:Math, then Template:Math. Because of this, the polynomial Template:Mvar is often also denoted by Template:Math. The image of the map Template:Tmath is denoted by Template:Math; it is the same thing as the subring of Template:Mvar generated by Template:Mvar and Template:Mvar.

Example: <math>k\left[t^2, t^3\right]</math> denotes the image of the homomorphism

<math>k[x, y] \to k[t], \, f \mapsto f\left(t^2, t^3\right).</math>

In other words, it is the subalgebra of Template:Math generated by Template:Math and Template:Math.

Example: let Template:Mvar be a polynomial in one variable, that is, an element in a polynomial ring Template:Mvar. Then Template:Math is an element in Template:Math and Template:Math is divisible by Template:Mvar in that ring. The result of substituting zero to Template:Mvar in Template:Math is Template:Math, the derivative of Template:Mvar at Template:Mvar.

The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism <math>\phi: R \to S</math> and an element Template:Mvar in Template:Mvar there exists a unique ring homomorphism <math>\overline{\phi}: R[t] \to S</math> such that <math>\overline{\phi}(t) = x</math> and <math>\overline{\phi}</math> restricts to Template:Mvar.Template:Sfnp For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring.

To give an example, let Template:Mvar be the ring of all functions from Template:Mvar to itself; the addition and the multiplication are those of functions. Let Template:Mvar be the identity function. Each Template:Mvar in Template:Mvar defines a constant function, giving rise to the homomorphism Template:Math. The universal property says that this map extends uniquely to

<math>R[t] \to S, \quad f \mapsto \overline{f}</math>

(Template:Mvar maps to Template:Mvar) where <math>\overline{f}</math> is the polynomial function defined by Template:Mvar. The resulting map is injective if and only if Template:Mvar is infinite.

Given a non-constant monic polynomial Template:Mvar in Template:Math, there exists a ring Template:Mvar containing Template:Mvar such that Template:Mvar is a product of linear factors in Template:Math.Template:Sfnp

Let Template:Mvar be an algebraically closed field. The Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in <math>k\left[t_1, \ldots, t_n\right]</math> and the set of closed subvarieties of Template:Mvar. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. Gröbner basis.)

There are some other related constructions. A formal power series ring <math>R[\![t]\!]</math> consists of formal power series

<math>\sum_0^\infty a_i t^i, \quad a_i \in R</math>

together with multiplication and addition that mimic those for convergent series. It contains Template:Math as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local (in fact, complete).

Matrix ring and endomorphism ringEdit

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Let Template:Mvar be a ring (not necessarily commutative). The set of all square matrices of size Template:Mvar with entries in Template:Mvar forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by Template:Math. Given a right Template:Mvar-module Template:Mvar, the set of all Template:Mvar-linear maps from Template:Mvar to itself forms a ring with addition that is of function and multiplication that is of composition of functions; it is called the endomorphism ring of Template:Mvar and is denoted by Template:Math.

As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: <math>\operatorname{End}_R(R^n) \simeq \operatorname{M}_n(R).</math> This is a special case of the following fact: If <math>f: \oplus_1^n U \to \oplus_1^n U</math> is an Template:Mvar-linear map, then Template:Mvar may be written as a matrix with entries Template:Mvar in Template:Math, resulting in the ring isomorphism:

<math>\operatorname{End}_R(\oplus_1^n U) \to \operatorname{M}_n(S), \quad f \mapsto (f_{ij}).</math>

Any ring homomorphism Template:Math induces Template:Math.Template:Sfnp

Schur's lemma says that if Template:Mvar is a simple right Template:Mvar-module, then Template:Math is a division ring.Template:Sfnp If <math>U = \bigoplus_{i = 1}^r U_i^{\oplus m_i}</math> is a direct sum of Template:Mvar-copies of simple Template:Mvar-modules <math>U_i,</math> then

<math>\operatorname{End}_R(U) \simeq \prod_{i=1}^r \operatorname{M}_{m_i} (\operatorname{End}_R(U_i)).</math>

The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form.

A ring Template:Mvar and the matrix ring Template:Math over it are Morita equivalent: the category of right modules of Template:Mvar is equivalent to the category of right modules over Template:Math.Template:Sfnp In particular, two-sided ideals in Template:Mvar correspond in one-to-one to two-sided ideals in Template:Math.

Limits and colimits of ringsEdit

Let Template:Mvar be a sequence of rings such that Template:Mvar is a subring of Template:Math for all Template:Mvar. Then the union (or filtered colimit) of Template:Mvar is the ring <math>\varinjlim R_i</math> defined as follows: it is the disjoint union of all Template:Mvar's modulo the equivalence relation Template:Math if and only if Template:Math in Template:Mvar for sufficiently large Template:Mvar.

Examples of colimits:

A polynomial ring in infinitely many variables: <math>R[t_1, t_2, \cdots] = \varinjlim R[t_1, t_2, \cdots, t_m].</math>
The algebraic closure of finite fields of the same characteristic <math>\overline{\mathbf{F}}_p = \varinjlim \mathbf{F}_{p^m}.</math>
The field of formal Laurent series over a field Template:Mvar: <math>k(\!(t)\!) = \varinjlim t^{-m}k[\![t]\!]</math> (it is the field of fractions of the formal power series ring <math>k[\![t]\!].</math>)
The function field of an algebraic variety over a field Template:Mvar is <math>\varinjlim k[U]</math> where the limit runs over all the coordinate rings Template:Math of nonempty open subsets Template:Mvar (more succinctly it is the stalk of the structure sheaf at the generic point.)

Any commutative ring is the colimit of finitely generated subrings.

A projective limit (or a filtered limit) of rings is defined as follows. Suppose we are given a family of rings Template:Math, Template:Math running over positive integers, say, and ring homomorphisms Template:Math, Template:Math such that Template:Math are all the identities and Template:Math is Template:Math whenever Template:Math. Then <math>\varprojlim R_i</math> is the subring of <math>\textstyle \prod R_i</math> consisting of Template:Math such that Template:Math maps to Template:Math under Template:Math, Template:Math.

For an example of a projective limit, see Template:Slink.

LocalizationEdit

The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring Template:Mvar and a subset Template:Mvar of Template:Mvar, there exists a ring <math>R[S^{-1}]</math> together with the ring homomorphism <math>R \to R\left[S^{-1}\right]</math> that "inverts" Template:Mvar; that is, the homomorphism maps elements in Template:Mvar to unit elements in <math>R\left[S^{-1}\right],</math> and, moreover, any ring homomorphism from Template:Mvar that "inverts" Template:Mvar uniquely factors through <math>R\left[S^{-1}\right].</math>Template:Sfnp The ring <math>R\left[S^{-1}\right]</math> is called the localization of Template:Mvar with respect to Template:Mvar. For example, if Template:Mvar is a commutative ring and Template:Mvar an element in Template:Mvar, then the localization <math>R\left[f^{-1}\right]</math> consists of elements of the form <math>r/f^n, \, r \in R , \, n \ge 0</math> (to be precise, <math>R\left[f^{-1}\right] = R[t]/(tf - 1).</math>)Template:Sfnp

The localization is frequently applied to a commutative ring Template:Mvar with respect to the complement of a prime ideal (or a union of prime ideals) in Template:Mvar. In that case <math>S = R - \mathfrak{p},</math> one often writes <math>R_\mathfrak{p}</math> for <math>R\left[S^{-1}\right].</math> <math>R_\mathfrak{p}</math> is then a local ring with the maximal ideal <math>\mathfrak{p} R_\mathfrak{p}.</math> This is the reason for the terminology "localization". The field of fractions of an integral domain Template:Mvar is the localization of Template:Mvar at the prime ideal zero. If <math>\mathfrak{p}</math> is a prime ideal of a commutative ring Template:Mvar, then the field of fractions of <math>R/\mathfrak{p}</math> is the same as the residue field of the local ring <math>R_\mathfrak{p}</math> and is denoted by <math>k(\mathfrak{p}).</math>

If Template:Mvar is a left Template:Mvar-module, then the localization of Template:Mvar with respect to Template:Mvar is given by a change of rings <math>M\left[S^{-1}\right] = R\left[S^{-1}\right] \otimes_R M.</math>

The most important properties of localization are the following: when Template:Mvar is a commutative ring and Template:Mvar a multiplicatively closed subset

<math>\mathfrak{p} \mapsto \mathfrak{p}\left[S^{-1}\right]</math> is a bijection between the set of all prime ideals in Template:Mvar disjoint from Template:Mvar and the set of all prime ideals in <math>R\left[S^{-1}\right].</math>Template:Sfnp
<math>R\left[S^{-1}\right] = \varinjlim R\left[f^{-1}\right],</math> Template:Mvar running over elements in Template:Mvar with partial ordering given by divisibility.Template:Sfnp
The localization is exact: <math display="block">0 \to M'\left[S^{-1}\right] \to M\left[S^{-1}\right] \to M\left[S^{-1}\right] \to 0</math> is exact over <math>R\left[S^{-1}\right]</math> whenever <math>0 \to M' \to M \to M \to 0</math> is exact over Template:Mvar.
Conversely, if <math>0 \to M'_\mathfrak{m} \to M_\mathfrak{m} \to M_\mathfrak{m} \to 0</math> is exact for any maximal ideal <math>\mathfrak{m},</math> then <math>0 \to M' \to M \to M \to 0</math> is exact.
A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)

In category theory, a localization of a category amounts to making some morphisms isomorphisms. An element in a commutative ring Template:Mvar may be thought of as an endomorphism of any Template:Mvar-module. Thus, categorically, a localization of Template:Mvar with respect to a subset Template:Mvar of Template:Mvar is a functor from the category of Template:Mvar-modules to itself that sends elements of Template:Mvar viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, Template:Mvar then maps to <math>R\left[S^{-1}\right]</math> and Template:Mvar-modules map to <math>R\left[S^{-1}\right]</math>-modules.)

CompletionEdit

Let Template:Mvar be a commutative ring, and let Template:Mvar be an ideal of Template:Mvar. The completion of Template:Mvar at Template:Mvar is the projective limit <math>\hat{R} = \varprojlim R/I^n;</math> it is a commutative ring. The canonical homomorphisms from Template:Mvar to the quotients <math>R/I^n</math> induce a homomorphism <math>R \to \hat{R}.</math> The latter homomorphism is injective if Template:Mvar is a Noetherian integral domain and Template:Mvar is a proper ideal, or if Template:Mvar is a Noetherian local ring with maximal ideal Template:Mvar, by Krull's intersection theorem.Template:Sfnp The construction is especially useful when Template:Mvar is a maximal ideal.

The basic example is the completion of Template:Tmath at the principal ideal Template:Math generated by a prime number Template:Mvar; it is called the ring of [[p-adic integer|Template:Mvar-adic integers]] and is denoted Template:Tmath The completion can in this case be constructed also from the [[p-adic absolute value|Template:Mvar-adic absolute value]] on Template:Tmath The Template:Mvar-adic absolute value on Template:Tmath is a map <math>x \mapsto |x|</math> from Template:Tmath to Template:Tmath given by <math>|n|_p=p^{-v_p(n)}</math> where <math>v_p(n)</math> denotes the exponent of Template:Mvar in the prime factorization of a nonzero integer Template:Mvar into prime numbers (we also put <math>|0|_p=0</math> and <math>|m/n|_p = |m|_p/|n|_p</math>). It defines a distance function on Template:Tmath and the completion of Template:Tmath as a metric space is denoted by Template:Tmath It is again a field since the field operations extend to the completion. The subring of Template:Tmath consisting of elements Template:Mvar with Template:Math is isomorphic to Template:Tmath

Similarly, the formal power series ring Template:Math is the completion of Template:Math at Template:Math (see also Hensel's lemma)

A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring.

Rings with generators and relationsEdit

The most general way to construct a ring is by specifying generators and relations. Let Template:Mvar be a free ring (that is, free algebra over the integers) with the set Template:Mvar of symbols, that is, Template:Mvar consists of polynomials with integral coefficients in noncommuting variables that are elements of Template:Mvar. A free ring satisfies the universal property: any function from the set Template:Mvar to a ring Template:Mvar factors through Template:Mvar so that Template:Math is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.Template:Sfnp

Now, we can impose relations among symbols in Template:Mvar by taking a quotient. Explicitly, if Template:Mvar is a subset of Template:Mvar, then the quotient ring of Template:Mvar by the ideal generated by Template:Mvar is called the ring with generators Template:Mvar and relations Template:Mvar. If we used a ring, say, Template:Mvar as a base ring instead of Template:Tmath then the resulting ring will be over Template:Mvar. For example, if <math>E = \{ xy - yx \mid x, y \in X \},</math> then the resulting ring will be the usual polynomial ring with coefficients in Template:Mvar in variables that are elements of Template:Mvar (It is also the same thing as the symmetric algebra over Template:Mvar with symbols Template:Mvar.)

In the category-theoretic terms, the formation <math>S \mapsto \text{the free ring generated by the set } S</math> is the left adjoint functor of the forgetful functor from the category of rings to Set (and it is often called the free ring functor.)

Let Template:Math, Template:Math be algebras over a commutative ring Template:Mvar. Then the tensor product of Template:Mvar-modules <math>A \otimes_R B</math> is an Template:Mvar-algebra with multiplication characterized by <math>(x \otimes u) (y \otimes v) = xy \otimes uv.</math> Template:See also

Special kinds of ringsEdit

DomainsEdit

A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.

Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra.Template:Sfnp Let Template:Mvar be a finite-dimensional vector space over a field Template:Mvar and Template:Math a linear map with minimal polynomial Template:Mvar. Then, since Template:Math is a unique factorization domain, Template:Mvar factors into powers of distinct irreducible polynomials (that is, prime elements): <math display="block">q = p_1^{e_1} \ldots p_s^{e_s}.</math>

Letting <math>t \cdot v = f(v),</math> we make Template:Mvar a Template:Math-module. The structure theorem then says Template:Mvar is a direct sum of cyclic modules, each of which is isomorphic to the module of the form <math>k[t] / \left(p_i^{k_j}\right).</math> Now, if <math>p_i(t) = t - \lambda_i,</math> then such a cyclic module (for Template:Mvar) has a basis in which the restriction of Template:Mvar is represented by a Jordan matrix. Thus, if, say, Template:Mvar is algebraically closed, then all Template:Mvar's are of the form Template:Math and the above decomposition corresponds to the Jordan canonical form of Template:Mvar.

File:Ringhierarchy.png

Hierarchy of several classes of rings with examples.

In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.Template:Sfnp

The following is a chain of class inclusions that describes the relationship between rings, domains and fields: Template:Commutative ring classes

Division ringEdit

A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).

Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.

The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.

A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.

Semisimple ringsEdit

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A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left module (or right module) over itself.

ExamplesEdit

A division ring is semisimple (and simple).
For any division ring Template:Mvar and positive integer Template:Mvar, the matrix ring Template:Math is semisimple (and simple).
For a field Template:Mvar and finite group Template:Mvar, the group ring Template:Math is semisimple if and only if the characteristic of Template:Mvar does not divide the order of Template:Mvar (Maschke's theorem).
Clifford algebras are semisimple.

The Weyl algebra over a field is a simple ring, but it is not semisimple. The same holds for a ring of differential operators in many variables.

PropertiesEdit

Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)

For a ring Template:Mvar, the following are equivalent:

Template:Mvar is semisimple.
Template:Mvar is artinian and semiprimitive.
Template:Mvar is a finite direct product <math display="inline"> \prod_{i=1}^r \operatorname{M}_{n_i}(D_i) </math> where each Template:Math is a positive integer, and each Template:Math is a division ring (Artin–Wedderburn theorem).

Semisimplicity is closely related to separability. A unital associative algebra Template:Mvar over a field Template:Mvar is said to be separable if the base extension <math>A \otimes_k F</math> is semisimple for every field extension Template:Math. If Template:Mvar happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.)

Central simple algebra and Brauer groupEdit

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For a field Template:Mvar, a Template:Mvar-algebra is central if its center is Template:Mvar and is simple if it is a simple ring. Since the center of a simple Template:Mvar-algebra is a field, any simple Template:Mvar-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a Template:Mvar-algebra. The matrix ring of size Template:Mvar over a ring Template:Mvar will be denoted by Template:Math.

The Skolem–Noether theorem states any automorphism of a central simple algebra is inner.

Two central simple algebras Template:Mvar and Template:Mvar are said to be similar if there are integers Template:Mvar and Template:Mvar such that <math>A \otimes_k k_n \approx B \otimes_k k_m.</math>Template:Sfnp Since <math>k_n \otimes_k k_m \simeq k_{nm},</math> the similarity is an equivalence relation. The similarity classes Template:Math with the multiplication <math>[A][B] = \left[A \otimes_k B\right]</math> form an abelian group called the Brauer group of Template:Mvar and is denoted by Template:Math. By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.

For example, Template:Math is trivial if Template:Mvar is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). <math>\operatorname{Br}(\R)</math> has order 2 (a special case of the theorem of Frobenius). Finally, if Template:Mvar is a nonarchimedean local field (for example, Template:Nowrap then <math>\operatorname{Br}(k) = \Q /\Z </math> through the invariant map.

Now, if Template:Mvar is a field extension of Template:Mvar, then the base extension <math>- \otimes_k F</math> induces Template:Math. Its kernel is denoted by Template:Math. It consists of Template:Math such that <math>A \otimes_k F</math> is a matrix ring over Template:Mvar (that is, Template:Mvar is split by Template:Mvar.) If the extension is finite and Galois, then Template:Math is canonically isomorphic to <math>H^2\left(\operatorname{Gal}(F/k), k^*\right).</math>Template:Sfnp

Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.

Valuation ringEdit

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If Template:Mvar is a field, a valuation Template:Mvar is a group homomorphism from the multiplicative group Template:Math to a totally ordered abelian group Template:Mvar such that, for any Template:Math, Template:Math in Template:Mvar with Template:Math nonzero, Template:Math The valuation ring of Template:Mvar is the subring of Template:Mvar consisting of zero and all nonzero Template:Mvar such that Template:Math.

Examples:

The field of formal Laurent series <math>k(\!(t)\!)</math> over a field Template:Mvar comes with the valuation Template:Mvar such that Template:Math is the least degree of a nonzero term in Template:Mvar; the valuation ring of Template:Mvar is the formal power series ring <math>k[\![t]\!].</math>
More generally, given a field Template:Mvar and a totally ordered abelian group Template:Mvar, let <math>k(\!(G)\!)</math> be the set of all functions from Template:Mvar to Template:Mvar whose supports (the sets of points at which the functions are nonzero) are well ordered. It is a field with the multiplication given by convolution: <math display="block">(f*g)(t) = \sum_{s \in G} f(s)g(t - s).</math> It also comes with the valuation Template:Mvar such that Template:Math is the least element in the support of Template:Mvar. The subring consisting of elements with finite support is called the group ring of Template:Mvar (which makes sense even if Template:Mvar is not commutative). If Template:Mvar is the ring of integers, then we recover the previous example (by identifying Template:Mvar with the series whose Template:Mvarth coefficient is Template:Math.)

Rings with extra structureEdit

A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:

An associative algebra is a ring that is also a vector space over a field Template:Mvar such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of Template:Mvar-by-Template:Mvar matrices over the real field Template:Tmath has dimension Template:Math as a real vector space.
A ring Template:Mvar is a topological ring if its set of elements Template:Mvar is given a topology which makes the addition map (<math>+ : R\times R \to R</math>) and the multiplication map Template:Math to be both continuous as maps between topological spaces (where Template:Math inherits the product topology or any other product in the category). For example, Template:Mvar-by-Template:Mvar matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
A λ-ring is a commutative ring Template:Mvar together with operations Template:Math that are like Template:Mvarth exterior powers:
<math>\lambda^n(x + y) = \sum_0^n \lambda^i(x) \lambda^{n-i}(y).</math>

For example, Template:Tmath is a λ-ring with <math>\lambda^n(x) = \binom{x}{n},</math> the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch theorem.

A totally ordered ring is a ring with a total ordering that is compatible with ring operations.

Some examples of the ubiquity of ringsEdit

Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.

Cohomology ring of a topological spaceEdit

To any topological space Template:Mvar one can associate its integral cohomology ring

<math>H^*(X,\Z ) = \bigoplus_{i=0}^{\infty} H^i(X,\Z ),</math>

a graded ring. There are also homology groups <math>H_i(X,\Z )</math> of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a Template:Mvar-multilinear form and an Template:Mvar-multilinear form to get a (Template:Math)-multilinear form.

The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.

Burnside ring of a groupEdit

To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.

Representation ring of a group ringEdit

To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.

Function field of an irreducible algebraic varietyEdit

To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.

Face ring of a simplicial complexEdit

Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.

Category-theoretic descriptionEdit

Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the [[tensor product of abelian groups|tensor product of Template:Tmath-modules]]). The monoid action of a ring Template:Mvar on an abelian group is simply an [[module (mathematics)|Template:Mvar-module]]. Essentially, an Template:Mvar-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".

Let Template:Math be an abelian group and let Template:Math be its endomorphism ring (see above). Note that, essentially, Template:Math is the set of all morphisms of Template:Mvar, where if Template:Mvar is in Template:Math, and Template:Mvar is in Template:Math, the following rules may be used to compute Template:Math and Template:Math:

<math>\begin{align}

& (f+g)(x) = f(x)+g(x) \\ & (f\cdot g)(x) = f(g(x)), \end{align}</math> where Template:Math as in Template:Math is addition in Template:Mvar, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, Template:Math, Template:Math is an abelian group. Furthermore, for every Template:Mvar in Template:Mvar, right (or left) multiplication by Template:Mvar gives rise to a morphism of Template:Math, by right (or left) distributivity. Let Template:Math. Consider those endomorphisms of Template:Mvar, that "factor through" right (or left) multiplication of Template:Mvar. In other words, let Template:Math be the set of all morphisms Template:Mvar of Template:Mvar, having the property that Template:Math. It was seen that every Template:Mvar in Template:Mvar gives rise to a morphism of Template:Mvar: right multiplication by Template:Mvar. It is in fact true that this association of any element of Template:Mvar, to a morphism of Template:Mvar, as a function from Template:Mvar to Template:Math, is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian Template:Mvar-group (by Template:Mvar-group, it is meant a group with Template:Mvar being its set of operators).Template:Sfnp In essence, the most general form of a ring, is the endomorphism group of some abelian Template:Mvar-group.

Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.

GeneralizationEdit

Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.

RngEdit

A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed.Template:Sfnp

Nonassociative ringEdit

A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.Template:Citation needed

SemiringEdit

A semiring (sometimes rig) is obtained by weakening the assumption that Template:Math is an abelian group to the assumption that Template:Math is a commutative monoid, and adding the axiom that Template:Math for all a in Template:Mvar (since it no longer follows from the other axioms).

Examples:

the non-negative integers <math>\{0,1,2,\ldots\}</math> with ordinary addition and multiplication;
the tropical semiring.

Other ring-like objectsEdit

Ring object in a categoryEdit

Let Template:Mvar be a category with finite products. Let pt denote a terminal object of Template:Mvar (an empty product). A ring object in Template:Mvar is an object Template:Mvar equipped with morphisms <math>R \times R\;\stackrel{a}\to\,R</math> (addition), <math>R \times R\;\stackrel{m}\to\,R</math> (multiplication), <math>\operatorname{pt}\stackrel{0}\to\,R</math> (additive identity), <math>R\;\stackrel{i}\to\,R</math> (additive inverse), and <math>\operatorname{pt}\stackrel{1}\to\,R</math> (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object Template:Mvar equipped with a factorization of its functor of points <math>h_R = \operatorname{Hom}(-,R) : C^{\operatorname{op}} \to \mathbf{Sets}</math> through the category of rings: <math>C^{\operatorname{op}} \to \mathbf{Rings} \stackrel{\textrm{forgetful}}\longrightarrow \mathbf{Sets}.</math>

Ring schemeEdit

In algebraic geometry, a ring scheme over a base scheme Template:Mvar is a ring object in the category of Template:Mvar-schemes. One example is the ring scheme Template:Math over Template:Tmath, which for any commutative ring Template:Mvar returns the ring Template:Math of Template:Mvar-isotypic Witt vectors of length Template:Mvar over Template:Mvar.<ref>Serre, p. 44</ref>

Ring spectrumEdit

In algebraic topology, a ring spectrum is a spectrum Template:Mvar together with a multiplication <math>\mu : X \wedge X \to X</math> and a unit map Template:Math from the sphere spectrum Template:Mvar, such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra.

NotesEdit

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CitationsEdit

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ReferencesEdit

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General referencesEdit

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Special referencesEdit

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Primary sourcesEdit

Historical referencesEdit

Bronshtein, I. N. and Semendyayev, K. A. (2004) Handbook of Mathematics, 4th ed. New York: Springer-Verlag Template:Isbn.
History of ring theory at the MacTutor Archive
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Faith, Carl (1999) Rings and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65. American Mathematical Society Template:Isbn.
Itô, K. editor (1986) "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press.
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