More formally, a ring is a set that is endowed with two binary operations (addition andmultiplication) such that the ring is anabelian group with respect to addition. The multiplication isassociative, isdistributive over the addition operation, and has a multiplicativeidentity element. Some authors apply the termring to a further generalization, often called arng, that omits the requirement for a multiplicative identity, and instead call the structure defined above aring with identity.
Acommutative ring is a ring with a commutative multiplication. This property has profound implications on ring properties.Commutative algebra, the theory of commutative rings, is a major branch ofring theory. Its development has been greatly influenced by problems and ideas ofalgebraic number theory andalgebraic geometry. In turn, commutative algebra is a fundaments tool in these branches of mathematics.
Aring is asetR equipped with twobinary operations[a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called thering axioms:[1][2][3]
In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a "rng" (IPA:/rʊŋ/) with a missing "i". For example, the set ofeven integers with the usual + and ⋅ is a rng, but not a ring. As explained in§ History below, many authors apply the term "ring" without requiring a multiplicative identity.
Although ring addition iscommutative, ring multiplication is not required to be commutative:ab need not necessarily equalba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are calledcommutative rings. Books on commutative algebra or algebraic geometry often adopt the convention thatring meanscommutative ring, to simplify terminology.
In a ring, multiplicative inverses are not required to exist. Anonzero commutative ring in which every nonzero element has amultiplicative inverse is called afield.
The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms.[4] The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products:ab +cd =cd +ab.)
There are a few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative.[5] For these authors, everyalgebra is a "ring".
The sum in is the remainder when the integerx +y is divided by4 (asx +y is always smaller than8, this remainder is eitherx +y orx +y − 4). For example, and
The product in is the remainder when the integerxy is divided by4. For example, and
Then is a ring: each axiom follows from the corresponding axiom for Ifx is an integer, the remainder ofx when divided by4 may be considered as an element of and this element is often denoted by "x mod 4" or which is consistent with the notation for0, 1, 2, 3. The additive inverse of any in is For example,
With the operations of matrix addition andmatrix multiplication, satisfies the above ring axioms. The element is the multiplicative identity of the ring. If and then while this example shows that the ring is noncommutative.
More generally, for any ringR, commutative or not, and any nonnegative integern, the squaren ×n matrices with entries inR form a ring; seeMatrix ring.
The study of rings originated from the theory ofpolynomial rings and the theory ofalgebraic integers.[11] In 1871,Richard Dedekind defined the concept of the ring of integers of a number field.[12] In this context, he introduced the terms "ideal" (inspired byErnst 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.
The term "Zahlring" (number ring) was coined byDavid Hilbert in 1892 and published in 1897.[13] 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 anequivalence).[14] 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, ifa3 − 4a + 1 = 0 then:
and so on; in general,an is going to be an integral linear combination of1,a, anda2.
The first axiomatic definition of a ring was given byAdolf Fraenkel in 1915,[15][16] but his axioms were stricter than those in the modern definition. For instance, he required everynon-zero-divisor to have amultiplicative inverse.[17] 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 paperIdealtheorie in Ringbereichen.[18]
Fraenkel applied the term "ring" to structures with axioms that included a multiplicative identity,[19] whereas Noether applied it to structures that did not.[18]
Most or all books on algebra[20][21] up to around 1960 followed Noether's convention of not requiring a1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of1 in the definition of "ring", especially in advanced books by notable authors such as Artin,[22] Bourbaki,[23] Eisenbud,[24] and Lang.[3] There are also books published as late as 2022 that use the term without the requirement for a1.[25][26][27][28] Likewise, theEncyclopedia of Mathematics does not require unit elements in rings.[29] 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 a1, 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."[30]Poonen makes the counterargument that the natural notion for rings would be thedirect 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.[c][31]
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",[32] or "ring with 1".[33]
to omit a requirement for a multiplicative identity: "rng"[34] or "pseudo-ring",[35] although the latter may be confusing because it also has other meanings.
The set of allcontinuous real-valuedfunctions defined on the real line forms a commutative-algebra. The operations arepointwise addition and multiplication of functions.
LetX be a set, and letR be a ring. Then the set of all functions fromX toR forms a ring, which is commutative ifR is commutative.
For any ringR and any natural numbern, the set of all squaren-by-nmatrices with entries fromR, forms a ring with matrix addition and matrix multiplication as operations. Forn = 1, this matrix ring is isomorphic toR itself. Forn > 1 (andR not the zero ring), this matrix ring is noncommutative.
IfG is anabelian group, then theendomorphisms ofG form a ring, theendomorphism ringEnd(G) of G. The operations in this ring are addition and composition of endomorphisms. More generally, ifV is aleft module over a ringR, then the set of allR-linear maps forms a ring, also called the endomorphism ring and denoted byEndR(V).
IfG is agroup andR is a ring, thegroup ring ofG overR is afree module overR havingG as basis. Multiplication is defined by the rules that the elements ofG commute with the elements ofR and multiply together as they do in the groupG.
The set ofnatural numbers with the usual operations is not a ring, since is not even agroup (not all the elements areinvertible with respect to addition – for instance, there is no natural number which can be added to3 to get0 as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers The natural numbers (including0) form an algebraic structure known as asemiring (which has all of the axioms of a ring excluding that of an additive inverse).
LetR 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 asconvolution: ThenR is a rng, but not a ring: theDirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of R.
A leftzero divisor of a ringR is an elementa in the ring such that there exists a nonzero elementb ofR such thatab = 0.[d] A right zero divisor is defined similarly.
Anilpotent element is an elementa such thatan = 0 for somen > 0. One example of a nilpotent element is anilpotent matrix. A nilpotent element in anonzero ring is necessarily a zero divisor.
Anidempotent is an element such thate2 =e. One example of an idempotent element is aprojection in linear algebra.
Aunit is an elementa having amultiplicative inverse; in this case the inverse is unique, and is denoted bya–1. The set of units of a ring is agroup under ring multiplication; this group is denoted byR× orR* orU(R). For example, ifR is the ring of all square matrices of sizen over a field, thenR× consists of the set of all invertible matrices of sizen, and is called thegeneral linear group.
A subsetS ofR is called asubring if any one of the following equivalent conditions holds:
the addition and multiplication ofRrestrict to give operationsS ×S →S makingS a ring with the same multiplicative identity as R.
1 ∈S; and for allx, y inS, the elementsxy,x +y, and−x are in S.
S can be equipped with operations making it a ring such that the inclusion mapS →R is a ring homomorphism.
For example, the ring of integers is a subring of thefield of real numbers and also a subring of the ring ofpolynomials (in both cases, contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers does not contain the identity element1 and thus does not qualify as a subring of one could call asubrng, however.
An intersection of subrings is a subring. Given a subsetE ofR, the smallest subring ofR containingE is the intersection of all subrings ofR containing E, and it is calledthe subring generated by E.
For a ringR, the smallest subring ofR is called thecharacteristic subring ofR. It can be generated through addition of copies of1 and −1. It is possible thatn · 1 = 1 + 1 + ... + 1 (n times) can be zero. Ifn is the smallest positive integer such that this occurs, thenn is called thecharacteristic of R. In some rings,n · 1 is never zero for any positive integern, and those rings are said to havecharacteristic zero.
Given a ringR, letZ(R) denote the set of all elementsx inR such thatx commutes with every element inR:xy =yx for anyy in R. ThenZ(R) is a subring of R, called thecenter of R. More generally, given a subsetX of R, letS be the set of all elements inR that commute with every element in X. ThenS is a subring of R, called thecentralizer (or commutant) of X. The center is the centralizer of the entire ring R. Elements or subsets of the center are said to becentral in R; they (each individually) generate a subring of the center.
LetR be a ring. Aleft ideal ofR is a nonempty subsetI ofR such that for anyx, y inI andr inR, the elementsx +y andrx are inI. IfR I denotes theR-span ofI, that is, the set of finite sums
thenI is a left ideal ifRI ⊆I. Similarly, aright ideal is a subsetI such thatIR ⊆I. A subsetI is said to be atwo-sided ideal or simplyideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup ofR. IfE is a subset ofR, thenRE is a left ideal, called the left ideal generated byE; it is the smallest left ideal containingE. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset ofR.
Ifx is inR, thenRx andxR are left ideals and right ideals, respectively; they are called theprincipal left ideals and right ideals generated byx. The principal idealRxR is written as(x). For example, the set of all positive and negative multiples of2 along with0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal.
Like a group, a ring is said to besimple 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 infinitechain of left ideals is called a leftNoetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a leftArtinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (theHopkins–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 idealP ofR is called aprime ideal if for any elements we have that implies either or Equivalently,P is prime if for any idealsI,J we have thatIJ ⊆P implies eitherI ⊆P orJ ⊆P. This latter formulation illustrates the idea of ideals as generalizations of elements.
Ahomomorphism from a ring(R, +,⋅) to a ring(S, ‡, ∗) is a functionf fromR to S that preserves the ring operations; namely, such that, for alla,b inR the following identities hold:
If one is working with rngs, then the third condition is dropped.
A ring homomorphismf is said to be anisomorphism if there exists an inverse homomorphism tof (that is, a ring homomorphism that is aninverse function), or equivalently if it isbijective.
Examples:
The function that maps each integerx to its remainder modulo4 (a number in{0, 1, 2, 3}) is a homomorphism from the ring to the quotient ring ("quotient ring" is defined below).
Ifu is a unit element in a ringR, then is a ring homomorphism, called aninner automorphism ofR.
LetR be a commutative ring of prime characteristicp. Thenx ↦xp is a ring endomorphism ofR called theFrobenius homomorphism.
TheGalois group of a field extensionL /K is the set of all automorphisms ofL whose restrictions toK are the identity.
For any ringR, there are a unique ring homomorphism and a unique ring homomorphismR → 0.
Anepimorphism (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map is an epimorphism.
Given a ring homomorphismf :R →S, the set of all elements mapped to 0 byf is called thekernel of f. The kernel is a two-sided ideal of R. The image of f, on the other hand, is not always an ideal, but it is always a subring of S.
To give a ring homomorphism from a commutative ringR to a ringA with image contained in the center ofA is the same as to give a structure of analgebra overR to A (which in particular gives a structure of anA-module).
The notion ofquotient ring is analogous to the notion of aquotient group. Given a ring(R, +,⋅) and a two-sidedidealI of(R, +,⋅), viewI as subgroup of(R, +); then thequotient ringR /I is the set ofcosets ofI together with the operations
for alla,b inR. The ringR /I is also called afactor ring.
As with a quotient group, there is a canonical homomorphismp :R →R /I, given byx ↦x +I. It is surjective and satisfies the following universal property:
Iff :R →S is a ring homomorphism such thatf(I) = 0, then there is a unique homomorphism such that
For any ring homomorphismf :R →S, invoking the universal property withI = kerf produces a homomorphism that gives an isomorphism fromR / kerf to the image off.
The concept of amodule over a ring generalizes the concept of avector space (over afield) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ringR, anR-moduleM is anabelian group equipped with anoperationR ×M →M (associating an element ofM to every pair of an element ofR and an element ofM) that satisfies certainaxioms. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for alla,b inR and allx,y inM,
M is an abelian group under addition.
When the ring isnoncommutative these axioms defineleft modules;right modules are defined similarly by writingxa instead ofax. This is not only a change of notation, as the last axiom of right modules (that isx(ab) = (xa)b) becomes(ab)x =b(ax), 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 (thedimension of a vector space). In particular, not all modules have abasis.
The axioms of modules imply that(−1)x = −x, where the first minus denotes theadditive 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: iff :R →S is a ring homomorphism, thenS is a left module overR by the multiplication:rs =f(r)s. IfR is commutative or iff(R) is contained in thecenter ofS, the ringS is called aR-algebra. In particular, every ring is an algebra over the integers.
LetR andS be rings. Then theproductR ×S can be equipped with the following natural ring structure:
for allr1,r2 inR ands1,s2 in S. The ringR ×S with the above operations of addition and multiplication and the multiplicative identity(1, 1) is called thedirect product ofR with S. The same construction also works for an arbitrary family of rings: ifRi are rings indexed by a setI, then is a ring with componentwise addition and multiplication.
LetR be a commutative ring and be ideals such that wheneveri ≠j. Then theChinese remainder theorem says there is a canonical ring isomorphism:
A "finite" direct product may also be viewed as a direct sum of ideals.[36] Namely, let be rings, the inclusions with the images (in particular are rings though not subrings). Then are ideals ofR andas 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 R. Equivalently, the above can be done throughcentral idempotents. Assume thatR has the above decomposition. Then we can writeBy the conditions on one has thatei are central idempotents andeiej = 0,i ≠j (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let which are two-sided ideals. If eachei is not a sum of orthogonal central idempotents,[e] then their direct sum is isomorphic to R.
An important application of an infinite direct product is the construction of aprojective limit of rings (see below). Another application is arestricted product of a family of rings (cf.adele ring).
Given a symbolt (called a variable) and a commutative ring R, the set of polynomials
forms a commutative ring with the usual addition and multiplication, containingR as a subring. It is called thepolynomial ring over R. More generally, the set of all polynomials in variables forms a commutative ring, containing as subrings.
IfR is anintegral domain, thenR[t] is also an integral domain; its field of fractions is the field ofrational functions. IfR is a Noetherian ring, thenR[t] is a Noetherian ring. IfR is a unique factorization domain, thenR[t] is a unique factorization domain. Finally,R is a field if and only ifR[t] is a principal ideal domain.
Let be commutative rings. Given an elementx of S, one can consider the ring homomorphism
(that is, thesubstitution). IfS =R[t] andx =t, thenf(t) =f. Because of this, the polynomialf is often also denoted byf(t). The image of the map is denoted byR[x]; it is the same thing as the subring ofS generated byR and x.
Example: denotes the image of the homomorphism
In other words, it is the subalgebra ofk[t] generated byt2 and t3.
Example: letf be a polynomial in one variable, that is, an element in a polynomial ringR. Thenf(x +h) is an element inR[h] andf(x +h) –f(x) is divisible byh in that ring. The result of substituting zero toh in(f(x +h) –f(x)) /h isf'(x), the derivative off at x.
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism and an elementx inS there exists a unique ring homomorphism such that and restricts toϕ.[37] For example, choosing a basis, asymmetric algebra satisfies the universal property and so is a polynomial ring.
To give an example, letS be the ring of all functions fromR to itself; the addition and the multiplication are those of functions. Letx be the identity function. Eachr inR defines a constant function, giving rise to the homomorphismR →S. The universal property says that this map extends uniquely to
(t maps tox) where is thepolynomial function defined byf. The resulting map is injective if and only ifR is infinite.
Given a non-constant monic polynomialf inR[t], there exists a ringS containingR such thatf is a product of linear factors inS[t].[38]
Letk be an algebraically closed field. TheHilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in and the set of closed subvarieties ofkn. 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. Aformal power series ring consists of formal power series
together with multiplication and addition that mimic those for convergent series. It containsR[t] 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 islocal (in fact,complete).
LetR be a ring (not necessarily commutative). The set of all square matrices of sizen with entries inR forms a ring with the entry-wise addition and the usual matrix multiplication. It is called thematrix ring and is denoted byMn(R). Given a rightR-moduleU, the set of allR-linear maps fromU to itself forms a ring with addition that is of function and multiplication that is ofcomposition of functions; it is called the endomorphism ring ofU and is denoted byEndR(U).
As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: This is a special case of the following fact: If is anR-linear map, thenf may be written as a matrix with entriesfij inS = EndR(U), resulting in the ring isomorphism:
Any ring homomorphismR →S inducesMn(R) → Mn(S).[39]
Schur's lemma says that ifU is a simple rightR-module, thenEndR(U) is a division ring.[40] If is a direct sum ofmi-copies of simpleR-modules then
A ringR and the matrix ringMn(R) over it areMorita equivalent: thecategory of right modules ofR is equivalent to the category of right modules overMn(R).[39] In particular, two-sided ideals inR correspond in one-to-one to two-sided ideals inMn(R).
LetRi be a sequence of rings such thatRi is a subring ofRi + 1 for alli. Then the union (orfiltered colimit) ofRi is the ring defined as follows: it is the disjoint union of allRi's modulo the equivalence relationx ~y if and only ifx =y inRi for sufficiently largei.
Aprojective limit (or afiltered limit) of rings is defined as follows. Suppose we are given a family of ringsRi,i running over positive integers, say, and ring homomorphismsRj →Ri,j ≥i such thatRi →Ri are all the identities andRk →Rj →Ri isRk →Ri wheneverk ≥j ≥i. Then is the subring of consisting of(xn) such thatxj maps toxi underRj →Ri,j ≥i.
For an example of a projective limit, see§ Completion.
Thelocalization generalizes the construction of thefield of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ringR and a subsetS ofR, there exists a ring together with the ring homomorphism that "inverts"S; that is, the homomorphism maps elements inS to unit elements in and, moreover, any ring homomorphism fromR that "inverts"S uniquely factors through[41] The ring is called thelocalization ofR with respect toS. For example, ifR is a commutative ring andf an element inR, then the localization consists of elements of the form (to be precise,)[42]
The localization is frequently applied to a commutative ringR with respect to the complement of a prime ideal (or a union of prime ideals) in R. In that case one often writes for is then alocal ring with themaximal ideal This is the reason for the terminology "localization". The field of fractions of an integral domainR is the localization ofR at the prime ideal zero. If is a prime ideal of a commutative ring R, then the field of fractions of is the same as the residue field of the local ring and is denoted by
IfM is a leftR-module, then the localization ofM with respect toS is given by achange of rings
The most important properties of localization are the following: whenR is a commutative ring andS a multiplicatively closed subset
is a bijection between the set of all prime ideals inR disjoint fromS and the set of all prime ideals in[43]
f running over elements inS with partial ordering given by divisibility.[44]
The localization is exact: is exact over whenever is exact over R.
Conversely, if is exact for any maximal ideal then 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.)
Incategory theory, alocalization of a category amounts to making some morphisms isomorphisms. An element in a commutative ringR may be thought of as an endomorphism of anyR-module. Thus, categorically, a localization ofR with respect to a subsetS ofR is afunctor from the category ofR-modules to itself that sends elements ofS viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course,R then maps to andR-modules map to-modules.)
LetR be a commutative ring, and letI be an ideal of R.Thecompletion ofR atI is the projective limit it is a commutative ring. The canonical homomorphisms fromR to the quotients induce a homomorphism The latter homomorphism is injective ifR is a Noetherian integral domain andI is a proper ideal, or ifR is a Noetherian local ring with maximal idealI, byKrull's intersection theorem.[45] The construction is especially useful whenI is a maximal ideal.
The basic example is the completion of at the principal ideal(p) generated by a prime numberp; it is called the ring ofp-adic integers and is denoted The completion can in this case be constructed also from thep-adic absolute value on Thep-adic absolute value on is a map from to given by where denotes the exponent ofp in the prime factorization of a nonzero integern into prime numbers (we also put and). It defines a distance function on and the completion of as ametric space is denoted by It is again a field since the field operations extend to the completion. The subring of consisting of elementsx with|x|p ≤ 1 is isomorphic to
Similarly, the formal power series ringR[{[t]}] is the completion ofR[t] at(t) (see alsoHensel's lemma)
A complete ring has much simpler structure than a commutative ring. This owns to theCohen 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 theintegral 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 ofexcellent ring.
The most general way to construct a ring is by specifying generators and relations. LetF be afree ring (that is, free algebra over the integers) with the setX of symbols, that is,F consists of polynomials with integral coefficients in noncommuting variables that are elements ofX. A free ring satisfies the universal property: any function from the setX to a ringR factors throughF so thatF →R is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.[46]
Now, we can impose relations among symbols inX by taking a quotient. Explicitly, ifE is a subset ofF, then the quotient ring ofF by the ideal generated byE is called the ring with generatorsX and relationsE. If we used a ring, say,A as a base ring instead of then the resulting ring will be overA. For example, if then the resulting ring will be the usual polynomial ring with coefficients inA in variables that are elements ofX (It is also the same thing as thesymmetric algebra overA with symbolsX.)
In the category-theoretic terms, the formation is the left adjoint functor of theforgetful functor from thecategory of rings toSet (and it is often called the free ring functor.)
LetA,B be algebras over a commutative ringR. Then the tensor product ofR-modules is anR-algebra with multiplication characterized by
Anonzero ring with no nonzerozero-divisors is called adomain. A commutative domain is called anintegral 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 aunique factorization domain (UFD), an integral domain in which every nonunit element is a product ofprime elements (an element is prime if it generates aprime ideal.) The fundamental question inalgebraic number theory is on the extent to which thering of (generalized) integers in anumber field, where an "ideal" admits prime factorization, fails to be a PID.
Among theorems concerning a PID, the most important one is thestructure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra.[47] LetV be a finite-dimensional vector space over a fieldk andf :V →V a linear map with minimal polynomialq. Then, sincek[t] is a unique factorization domain,q factors into powers of distinct irreducible polynomials (that is, prime elements):
Letting we makeV ak[t]-module. The structure theorem then saysV is a direct sum ofcyclic modules, each of which is isomorphic to the module of the form Now, if then such a cyclic module (forpi) has a basis in which the restriction off is represented by aJordan matrix. Thus, if, say,k is algebraically closed, then allpi's are of the formt –λi and the above decomposition corresponds to theJordan canonical form off.
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 aregular local ring. A regular local ring is a UFD.[48]
The following is a chain ofclass inclusions that describes the relationship between rings, domains and fields:
Adivision ring is a ring such that every non-zero element is a unit. A commutative division ring is afield. A prominent example of a division ring that is not a field is the ring ofquaternions. 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 everyfinite domain (in particular finite division ring) is a field; in particular commutative (theWedderburn'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, theCartan–Brauer–Hua theorem.
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.)
Semisimplicity is closely related to separability. A unital associative algebraA over a fieldk is said to beseparable if the base extension is semisimple for everyfield extensionF /k. IfA happens to be a field, then this is equivalent to the usual definition in field theory (cf.separable extension.)
For a fieldk, ak-algebra is central if its center isk and is simple if it is asimple ring. Since the center of a simplek-algebra is a field, any simplek-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 ak-algebra. The matrix ring of sizen over a ringR will be denoted byRn.
Two central simple algebrasA andB are said to besimilar if there are integersn andm such that[49] Since the similarity is an equivalence relation. The similarity classes[A] with the multiplication form an abelian group called theBrauer group ofk and is denoted byBr(k). By theArtin–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.
Now, ifF is a field extension ofk, then the base extension inducesBr(k) → Br(F). Its kernel is denoted byBr(F /k). It consists of[A] such that is a matrix ring overF (that is,A is split byF.) If the extension is finite and Galois, thenBr(F /k) is canonically isomorphic to[50]
Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.
IfK is a field, avaluationv is a group homomorphism from the multiplicative groupK∗ to a totally ordered abelian groupG such that, for anyf,g inK withf +g nonzero,v(f +g) ≥ min{v(f),v(g)}. Thevaluation ring ofv is the subring ofK consisting of zero and all nonzerof such thatv(f) ≥ 0.
Examples:
The field offormal Laurent series over a fieldk comes with the valuationv such thatv(f) is the least degree of a nonzero term inf; the valuation ring ofv is theformal power series ring
More generally, given a fieldk and a totally ordered abelian groupG, let be the set of all functions fromG tok whose supports (the sets of points at which the functions are nonzero) arewell ordered. It is a field with the multiplication given byconvolution: It also comes with the valuationv such thatv(f) is the least element in the support off. The subring consisting of elements with finite support is called thegroup ring ofG (which makes sense even ifG is not commutative). IfG is the ring of integers, then we recover the previous example (by identifyingf with the series whosenth coefficient is f(n).)
A ring may be viewed as anabelian 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:
Anassociative algebra is a ring that is also avector space over a fieldn such that the scalar multiplication is compatible with the ring multiplication. For instance, the set ofn-by-n matrices over the real field has dimensionn2 as a real vector space.
A ringR is atopological ring if its set of elementsR is given atopology which makes the addition map () and the multiplication map⋅ :R ×R →R to be bothcontinuous as maps between topological spaces (whereX ×X inherits theproduct topology or any other product in the category). For example,n-by-n matrices over the real numbers could be given either theEuclidean topology, or theZariski topology, and in either case one would obtain a topological ring.
Aλ-ring is a commutative ringR together with operationsλn:R →R that are likenthexterior powers:
agraded ring. There are alsohomology groups of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like thespheres andtori, for which the methods ofpoint-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 avector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of theuniversal coefficient theorem. However, the advantage of the cohomology groups is that there is anatural product, which is analogous to the observation that one can multiply pointwise ak-multilinear form and anl-multilinear form to get a (k +l)-multilinear form.
To anygroup is associated itsBurnside ring which uses a ring to describe the various ways the group canact on a finite set. The Burnside ring's additive group is thefree 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 therepresentation 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.
To anygroup ring orHopf algebra is associated itsrepresentation 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 fromcharacter theory, which is more or less theGrothendieck group given a ring structure.
Function field of an irreducible algebraic variety
Everysimplicial complex has an associated face ring, also called itsStanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest inalgebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension ofsimplicial polytopes.
Let(A, +) be an abelian group and letEnd(A) be itsendomorphism ring (see above). Note that, essentially,End(A) is the set of all morphisms ofA, where iff is inEnd(A), andg is inEnd(A), the following rules may be used to computef +g andf ⋅g:
where+ as inf(x) +g(x) is addition inA, and function composition is denoted from right to left. Therefore,associated to any abelian group, is a ring. Conversely, given any ring,(R, +,⋅ ),(R, +) is an abelian group. Furthermore, for everyr inR, right (or left) multiplication byr gives rise to a morphism of(R, +), by right (or left) distributivity. LetA = (R, +). Consider thoseendomorphisms ofA, that "factor through" right (or left) multiplication ofR. In other words, letEndR(A) be the set of all morphismsm ofA, having the property thatm(r ⋅x) =r ⋅m(x). It was seen that everyr inR gives rise to a morphism ofA: right multiplication byr. It is in fact true that this association of any element ofR, to a morphism ofA, as a function fromR toEndR(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelianX-group (byX-group, it is meant a group withX being itsset of operators).[51] In essence, the most general form of a ring, is the endomorphism group of some abelianX-group.
Any ring can be seen as apreadditive 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 ofmorphisms closed under addition and under composition with arbitrary morphisms.
Anonassociative 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 aLie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.[citation needed]
Asemiring (sometimesrig) is obtained by weakening the assumption that(R, +) is an abelian group to the assumption that(R, +) is a commutative monoid, and adding the axiom that0 ⋅a =a ⋅ 0 = 0 for alla inR (since it no longer follows from the other axioms).
Examples:
the non-negative integers with ordinary addition and multiplication;
LetC be a category with finiteproducts. Let pt denote aterminal object ofC (an empty product). Aring object inC is an objectR equipped with morphisms (addition), (multiplication), (additive identity), (additive inverse), and (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an objectR equipped with a factorization of its functor of points through the category of rings:
In algebraic geometry, aring scheme over a baseschemeS is a ring object in the category ofS-schemes. One example is the ring schemeWn over, which for any commutative ringA returns the ringWn(A) ofp-isotypicWitt vectors of lengthn overA.[53]
^This means that each operation is defined and produces a unique result inR for each ordered pair of elements ofR.
^The existence of 1 is not assumed by some authors; here, the termrng is used if existence of a multiplicative identity is not assumed. Seenext subsection.
^Poonen claims that "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a 1".
^Some other authors such as Lang further require a zero divisor to be nonzero.
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Balcerzyk, Stanisław; Józefiak, Tadeusz (1989),Commutative Noetherian and Krull rings, Mathematics and its Applications, Chichester: Ellis Horwood Ltd.,ISBN978-0-13-155615-7
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