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In ring theory, a branch of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
in
group theory In 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 ...
and to the quotient space in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
. It is a specific example of a quotient, as viewed from the general setting of
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
. Starting with a ring and a two-sided ideal in , a new ring, the quotient ring , is constructed, whose elements are the
cosets In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
of in subject to special and operations. (Only the fraction slash "/" is used in quotient ring notation, not a horizontal fraction bar.) 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.


Formal quotient ring construction

Given a ring and a two-sided ideal in , we may define an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
on as follows: :
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
is in . Using the ideal properties, it is not difficult to check that is a
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
. In case , we say that and are ''congruent
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
'' . The
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of the element in is given by : a+I := \. This equivalence class is also sometimes written as a \bmod I and called the "residue class of modulo ". The set of all such equivalence classes is denoted by ; it becomes a ring, the factor ring or quotient ring of modulo , if one defines :\begin & (a+I)+(b+I)=(a+b)+I; \\ & (a+I)(b+I)=(ab)+I. \end (Here one has to check that these definitions are well-defined. Compare coset and
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
.) The zero-element of is \bar=(0+I)=I, and the multiplicative identity is \bar = (1+I). The map from to defined by p(a)=a+I is a
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
ring homomorphism, sometimes called the ''natural quotient map'' or the ''
canonical homomorphism In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A ...
''.


Examples

*The quotient ring is
naturally isomorphic In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to , and is the zero ring since, by our definition, for any in , we have that = r+R := \, which equals itself. This fits with the rule of thumb that the larger the ideal , the smaller the quotient ring . If is a proper ideal of , i.e., , then is not the zero ring. *Consider the ring of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s and the ideal of even numbers, denoted by Then the quotient ring has only two elements, the coset consisting of the even numbers and the coset consisting of the odd numbers; applying the definition, = z+2\Z := \, where is the ideal of even numbers. It is naturally isomorphic to the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with two 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 In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
is essentially arithmetic in the quotient ring (which has elements). *Now consider the ring of polynomials in the variable with real
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s, and the ideal I=(X^2+1) consisting of all multiples of the
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
X^2+1. The quotient ring is naturally isomorphic to the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s with the class playing the role of the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. The reason is that we "forced" X^2+1=0, i.e. X^2=-1, which is the defining property of . *Generalizing the previous example, quotient rings are often used to construct
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s. Suppose is some field and is an irreducible polynomial in . Then is a field whose minimal polynomial over is , which contains as well as an element . *One important instance of the previous example is the construction of the finite fields. Consider for instance the field \mathbb F_3 = \Z / 3\Z with three elements. The polynomial f(X)=X^2+1 is irreducible over (since it has no root), and we can construct the quotient ring This is a field with elements, denoted by 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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. As a simple case, consider the real variety V = \ as a subset of the real plane The ring of real-valued polynomial functions defined on can be identified with the quotient ring and this is the coordinate ring of . The variety is now investigated by studying its coordinate ring. *Suppose is a -
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, and is a point of . Consider the ring of all -functions defined on and let be the ideal in consisting of those functions which are identically zero in some
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of (where may depend on ). Then the quotient ring is the ring of germs of -functions on at . *Consider the ring of finite elements of a hyperreal field It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers for which a standard integer with exists. The set of all infinitesimal numbers in together with 0, is an ideal in , and the quotient ring is isomorphic to the real numbers The isomorphism is induced by associating to every element of the standard part of , i.e. the unique real number that differs from by an infinitesimal. In fact, one obtains the same result, namely if one starts with the ring of finite hyperrationals (i.e. ratio of a pair of hyperintegers), see construction of the real numbers.


Variations of complex planes

The quotients and are all isomorphic to and gain little interest at first. But note that is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of by This variation of a complex plane arises as a subalgebra whenever the algebra contains a
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
and a
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
. Furthermore, the ring quotient does split into and so this ring is often viewed as the direct sum Nevertheless, a variation on complex numbers z=x+yj is suggested by as a root of X^2-1, compared to as root of X^2+1=0. This plane of
split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s normalizes the direct sum by providing a basis \ for 2-space where the identity of the algebra is at unit distance from the zero. With this basis a
unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...
may be compared to the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
of the ordinary complex plane.


Quaternions and variations

Suppose and are two, non-commuting, indeterminates and form the free algebra Then Hamilton’s
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s of 1843 can be cast as :\R \langle X,Y \rangle / ( X^2+1, Y^2+1, XY+YX) . If is substituted for then one obtains the ring of split-quaternions. The anti-commutative property implies that has as its square :(XY)(XY) = X(YX)Y = -X(XY)Y = -(XX)(YY) = -(-1)(+1) = +1. Substituting minus for plus in ''both'' the quadratic binomials also results in split-quaternions. The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminates and constructing appropriate ideals.


Properties

Clearly, if is a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, then so is ; the converse, however, is not true in general. The natural quotient map has 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. The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: :the ring homomorphisms defined on are essentially the same as the ring homomorphisms defined on that vanish (i.e. are zero) on . More precisely, given a two-sided ideal in and a ring homomorphism whose kernel contains , there exists precisely one ring homomorphism with (where is the natural quotient map). The map here is given by the well-defined rule for all in . Indeed, this
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
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 induces a ring isomorphism between the quotient ring and the image . (See also: fundamental theorem on homomorphisms.) The ideals of and are closely related: the natural quotient map provides a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the two-sided ideals of that contain and the two-sided ideals of (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 is a two-sided ideal in that contains , and we write for the corresponding ideal in (i.e. ), the quotient rings and are naturally isomorphic via the (well-defined!) mapping . The following facts prove useful in
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
: for commutative, is a field if and only if is a maximal ideal, while is an integral domain if and only if is a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
. A number of similar statements relate properties of the ideal to properties of the quotient ring . The
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
states that, if the ideal is the intersection (or equivalently, the product) of pairwise coprime ideals , then the quotient ring is isomorphic to the product of the quotient rings , .


For algebras over a ring

An associative algebra over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
  is a ring itself. If is an ideal in  (closed under -multiplication), then inherits the structure of an algebra over  and is the quotient algebra.


See also

* Associated graded ring * Residue field *
Goldie's theorem In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring ''R'' that has finite uniform dimension (="finite rank") as a right module ...
* Quotient module


Notes


Further references

* F. Kasch (1978) ''Moduln und Ringe'', translated by DAR Wallace (1982) ''Modules and Rings'',
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes refer ...
, page 33. * Neal H. McCoy (1948) ''Rings and Ideals'', §13 Residue class rings, page 61, Carus Mathematical Monographs #8,
Mathematical Association of America The Mathematical Association of America (MAA) 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 a ...
. * * 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

* {{springer, title=Quotient ring, id=p/q076920
Ideals and factor rings
from John Beachy's ''Abstract Algebra Online'' Ring Ring theory