Quotient ring
Encyclopedia
In ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...

, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

 and the quotient space
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....

s of linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

. One starts with a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 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. 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
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...

, of an integral domain as well as from the more general 'rings of quotients' obtained by localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

.

Formal quotient ring construction

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

 ~ on R as follows:
a ~ b if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 b − a is in I.

Using the ideal properties, it is not difficult to check that ~ is a congruence relation
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

.
In case a ~ b, we say that a and b are congruent modulo
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

 I.
The equivalence class of the element a in R is given by
[a] = a + I := { a + r : r in I }.


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) = (a b) + I.

(Here one has to check that these definitions are well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...

. Compare coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...

 and quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

.) 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
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....

, sometimes called the natural quotient map or the canonical homomorphism.

Examples

  • The most extreme examples of quotient rings are provided by modding out
    Modulo (jargon)
    The word modulo is the Latin ablative of modulus which itself means "a small measure."It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801...

     the most extreme ideals, {0} and R itself. R/{0} is naturally isomorphic to R, and R/R is the trivial ring {0}. 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 integer
    Integer
    The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

    s Z and the ideal of even numbers, denoted by 2Z. Then the quotient ring Z/2Z has only two elements, zero for the even numbers and one for the odd numbers. It is naturally isomorphic to the finite field
    Finite field
    In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

     with two elements, F2. 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
    Modular arithmetic
    In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

     is essentially arithmetic in the quotient ring Z/nZ (which has n elements).

  • Now consider the ring R[X] of polynomial
    Polynomial ring
    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 with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

    s in the variable X with real
    Real number
    In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

     coefficients, and the ideal I = (X2 + 1) consisting of all multiples of the polynomial X2 + 1. The quotient ring R[X]/(X2 + 1) is naturally isomorphic to the field of complex number
    Complex number
    A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

    s C, with the class [X] playing the role of the imaginary unit
    Imaginary unit
    In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

     i. 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 extension
    Field extension
    In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

    s. Suppose K is some field
    Field (mathematics)
    In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

     and f is an irreducible polynomial
    Irreducible polynomial
    In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....

     in K[X]. Then L = K[X]/(f) is a field whose minimal polynomial
    Minimal polynomial (field theory)
    In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...

     over K is f, which contains K as well as an element x = X + (f).

  • One important instance of the previous example is the construction of the finite fields. 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
    Algebraic variety
    In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

     are important examples of quotient rings in algebraic geometry
    Algebraic geometry
    Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

    . 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
    Manifold
    In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

    , and p is a point of M. 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). Then the quotient ring R/I is the ring of germ
    Germ (mathematics)
    In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...

    s of C-functions on M at p.

  • Consider the ring F of finite elements of a hyperreal field
    Hyperreal number
    The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...

     *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 R. The isomorphism is induced by associating to every element x of F the standard part
    Standard part function
    In non-standard analysis, the standard part function is a function from the limited hyperreals to the reals, which associates to every hyperreal, the unique real infinitely close to it. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de...

     of x, i.e. the unique real number that differs from x by an infinitesimal. In fact, one obtains the same result, namely R, if one starts with the ring F of finite hyperrationals (i.e. ratio of a pair of hyperinteger
    Hyperinteger
    In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer...

    s), see construction of the real numbers.

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
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...

 plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of R[X] by X2. This alternative complex plane arises as a subalgebra
Subalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...

 whenever the algebra contains a real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

 and a nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....

.

Furthermore, the ring quotient R[X]/(X2 − 1) does split into R[X]/(X + 1) and R[X]/(X − 1), so this ring is often viewed as the direct sum R  R.
Nevertheless, an alternative complex number z = x + y j is suggested by j as a root of X2 − 1, compared to i as root of X2 + 1 = 0. This plane of split-complex number
Split-complex number
In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...

s normalizes the direct sum by providing a basis {1, j } for 2-space where the identity of the algebra is at unit distance from the zero. With this basis a unit hyperbola
Unit hyperbola
In geometry, the unit hyperbola is the set of points in the Cartesian plane that satisfies x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial lengthWhereas the unit circle surrounds its center, the unit hyperbola requires the...

 may be compared to the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

 of the ordinary complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

.

Quaternions and alternatives

Hamilton’s quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s of 1843 can be cast as R[X,Y]/(X2 + 1, Y2 + 1, XY + YX). If Y2 − 1 is substituted for Y2 + 1, then one obtains the ring of split-quaternions. 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.


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.

Properties

Clearly, if R is a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

, then so is R/I; the converse however is not true in general.

The natural quotient map p has I as its kernel
Kernel (algebra)
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 fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

; 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.

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
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...

 can be used to define quotient rings and their natural quotient maps.

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). (See also: fundamental theorem on homomorphisms
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 between which a homomorphism is given, and of the kernel and image of the homomorphism....

.)

The ideals of R and R/I are closely related: the natural quotient map provides a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

 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). 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
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

 and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, the following statement is often used: If R ≠ {0} is a commutative ring and I is a maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...

, then the quotient ring R/I is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

; if I is only a prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

, then R/I is only an integral domain. A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I.

The Chinese remainder theorem
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...

 states that, if the ideal I is the intersection (or equivalently, the product) of pairwise coprime ideals I1,...,Ik, then the quotient ring R/I is isomorphic to the product
Product of rings
In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e...

 of the quotient rings R/Ip , p=1,...,k.

Further References

  • F. Kasch (1978) Moduln und Ringe, translated by DAR Wallace (1982) Modules and Rings, Academic Press
    Academic Press
    Academic Press is an academic book publisher. Originally independent, it was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier....

    , page 33.
  • Neal H. McCoy (1948) Rings and Ideals, §13 Residue class rings, page 61, Carus Mathematical Monographs #8, Mathematical Association of America
    Mathematical Association of America
    The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...

    .
  • B.L. van der Waerden (1970) Algebra, translated by Fred Blum and John R Schulenberger, Frederick Ungar Publishing, New York. See Chapter 3.5, "Ideals. Residue Class Rings", pages 47 to 51.

External links

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