In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N Linear algebra is the branch of Mathematics concerned with One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I, essentially by requiring that all elements of I be zero. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Intuitively, the quotient ring R/I is a "simplified version" of R where the elements of I are "ignored".
Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well as from the more general 'rings of quotients' obtained by localization. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring.
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Formal quotient ring construction
Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:
- a ~ b if and only if b − a is in I. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" ↔
Using the ideal properties, it is not difficult to check that ~ is a congruence relation. See Congruence (geometry for the term as used in elementary geometry In case a ~ b, we say that a and b are congruent modulo I. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure The equivalence class of the element a in R is given by
- [a] = a + I := { a + r : r in I }. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X
This equivalence class is also sometimes written as a mod I and called the "residue class of a modulo I".
The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of R modulo I, if one defines
- (a + I) + (b + I) = (a + b) + I;
- (a + I)(b + I) = (ab) + I.
(Here one has to check that these definitions are well-defined. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc Compare coset and quotient group. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G ) The zero-element of R/I is (0 + I) = I, and the multiplicative identity is (1 + I).
The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication
Examples
The most extreme examples of quotient rings are provided by modding out the most extreme ideals, {0} and R itself. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure R/{0} is naturally isomorphic to R, and R/R is the trivial ring {0}. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal A trivial ring is a ring defined on a Singleton set, { r } The ring operations (× and + are trivial r \times r = r This fits with the general rule of thumb that the smaller the ideal I, the larger the quotient ring R/I. If I is a proper ideal of R, i. e. I ≠ R, then R/I won't be the trivial ring.
Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the parity of an object states whether it is even or odd Then the quotient ring Z/2Z has only two elements, one for the even numbers and one for the odd numbers. It is naturally isomorphic to the finite field with two elements, F2. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements Intuitively: if you think of all the even numbers as 0, then every integer is either 0 (if it is even) or 1 (if it is odd and therefore differs from an even number by 1). Modular arithmetic is essentially arithmetic in the quotient ring Z/nZ (which has n elements). In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers
Now consider the ring R[X] of polynomials in the variable X with real coefficients, and the ideal I = (X2 + 1) consisting of all multiples of the polynomial X2 + 1. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables In Mathematics, the real numbers may be described informally in several different ways The quotient ring R[X]/(X2 + 1) is naturally isomorphic to the field of complex numbers C, with the class [X] playing the role of the imaginary unit i. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation The reason: we "forced" X2 + 1 = 0, i. e. X2 = -1, which is the defining property of i.
Generalizing the previous example, quotient rings are often used to construct field extensions. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. Suppose K is some field and f is an irreducible polynomial in K[X]. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given set Then L = K[X]/(f) is a field which contains K as well as an element x = X + (f) whose minimal polynomial over K is f. In Linear algebra, the minimal polynomial of an n -by- n matrix A over a field F is the Monic polynomial
One important instance of the previous example is the construction of the finite fields. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements Consider for instance the field F3 = Z/3Z with three elements. The polynomial f(X) = X2 + 1 is irreducible over F3 (since it has no root), and we can construct the quotient ring F3[X]/(f). This is a field with 32=9 elements, denoted by F9. The other finite fields can be constructed in a similar fashion.
The coordinate rings of algebraic varieties are important examples of quotient rings in algebraic geometry. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with As a simple case, consider the real variety V = {(x,y) | x2 = y3 } as a subset of the real plane R2. The ring of real-valued polynomial functions defined on V can be identified with the quotient ring R[X,Y]/(X2 - Y3), and this is the coordinate ring of V. The variety V is now investigated by studying its coordinate ring.
Suppose M is a C∞ -manifold, and p is a point of M. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Consider the ring R=C∞(M) of all C∞-functions defined on M and let I be the ideal in R consisting of those functions f which are identically zero in some neighborhood U of p (where U may depend on f). In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Then the quotient ring R/I is the ring of germs of C∞-functions on M at p. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable
Consider the ring F of finite elements of a hyperreal field *R. It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers x for which a standard integer n with -n < x < n exists. The set I of all infinitesimal numbers in *R, together with 0, is an ideal in F, and the quotient ring F/I is isomorphic to the real numbers. The isomorphism is induced by associating to every element x of F the standard part of x, i. e. the unique real number that differs from x by an infinitesimal.
Alternative complex planes
The quotients R[X]/(x) , R[X]/(x+1), and R[X]/(x-1) are all isomorphic to R and gain little interest at first. But note that R[X]/(X2) is called the dual number plane in geometric algebra. A variety of dualities in mathematics are listed at Duality (mathematics. It consists only of linear binomials as “remainders” after reducing an element of R[X] by X2. This alternative complex plane arises frequently enough to accent its existence.
Furthermore, the ring quotient R[X]/(X2 −1) does split into R[X]/(X+1) and R[X]/(X−1), so this split-complex number ring is often viewed as the direct sum R 
R. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction Nevertheless, a complex number structure based on a hyperbola is brought in. The planar linear algebra of squeeze mapping , a. In Linear algebra, a squeeze mapping is a type of Linear map that preserves Euclidean Area of regions in the Cartesian plane, but is not a k. a. hyperbolic rotation, fits naturally. The parallel with ordinary complex number representation of circular rotation is a part of split-complex number assignments and arithmetic.
Quaternions and alternatives
Hamilton’s quaternions of 1843 can be cast as R[X,Y]/(X2 + 1, Y2 + 1, XY + YX). Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician If Y2 − 1 is substituted for Y2 + 1, then one obtains the ring of split-quaternions. In Abstract algebra, the split-quaternions or coquaternions are elements of an Associative algebra introduced by James Cockle in 1849 under the Substituting minus for plus in both the quadratic binomials also results in split-quaternions: The anti-commutative property YX = −XY implies that XY has for its square
- (XY)(XY) = X(YX)X = − X(XY)Y = − XXYY = − 1. In mathematics anticommutativity refers to the property of an operation being anticommutative, i
The three types of biquaternions can also be written as quotients by conscripting the three-indeterminate ring R[X,Y,Z] and constructing appropriate ideals. The biquaternions are the numbers w + xi + yj + zk \ \! where w x y and z are complex numbers and the elements of {1 i j k} multiply as in the Quaternion group
Properties
Clearly, if R is a commutative ring, then so is R/I; the converse however is not true in general. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property
The natural quotient map p has I as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism
The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: the ring homomorphisms defined on R/I are essentially the same as the ring homomorphisms defined on R that vanish (i. e. are zero) on I. More precisely: given a two-sided ideal I in R and a ring homomorphism f : R → S whose kernel contains I, then there exists precisely one ring homomorphism g : R/I → S with gp = f (where p is the natural quotient map). The map g here is given by the well-defined rule g([a]) = f(a) for all a in R. Indeed, this universal property can be used to define quotient rings and their natural quotient maps. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism
As a consequence of the above, one obtains the fundamental statement: every ring homomorphism f : R → S induces a ring isomorphism between the quotient ring R/ker(f) and the image im(f). In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication (See also: fundamental theorem on homomorphisms. In Abstract algebra, the Fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects )
The ideals of R and R/I are closely related: the natural quotient map provides a bijection between the two-sided ideals of R that contain I and the two-sided ideals of R/I (the same is true for left and for right ideals). In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if M is a two-sided ideal in R that contains I, and we write M/I for the corresponding ideal in R/I (i. e. M/I = p(M)), the quotient rings R/M and (R/I)/(M/I) are naturally isomorphic via the (well-defined!) mapping a + M |-> (a+I) + M/I.
In commutative algebra and algebraic geometry, the following statement is often used: If R ≠ {0} is a commutative ring and I is a maximal ideal, then the quotient ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I.
The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwise coprime ideals I1,. The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra. . . ,Ik, then the quotient ring R/I is isomorphic to the product of the quotient rings R/Ip , p=1,. In Mathematics, it is possible to combine several rings into one large product ring. . . ,k.
See also
External links
- Ideals and factor rings from John Beachy's Abstract Algebra Online
- Quotient ring on PlanetMath
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