In
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, a branch of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the
quotient group in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
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 matrices ...
.
It is a specific example of a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
, 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 stu ...
. Starting with a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
and a
two-sided ideal
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers p ...
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
The slash is the oblique slanting line punctuation mark . Also known as a stroke, a solidus or several other historical or technical names including oblique and virgule. Once used to mark periods and commas, the slash is now used to represen ...
"/" is used in quotient ring notation, not a horizontal
fraction bar
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
.)
Quotient rings are distinct from the so-called "quotient field", or
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
, of an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
as well as from the more general "rings of quotients" obtained by
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
.
Formal quotient ring construction
Given a ring and a two-sided ideal in , we may define an
equivalence relation 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 b ...
is in .
Using the ideal properties, it is not difficult to check that is a
congruence relation.
In case , we say that and are ''congruent
modulo'' .
The
equivalence class of the element in is given by
:
This equivalence class is also sometimes written as
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
:
(Here one has to check that these definitions are
well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
. Compare
coset
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) ...
and
quotient group.) The zero-element of is
and the multiplicative identity is
The map from to defined by
is a
surjective ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
, 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 natu ...
to , and is the
zero ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
since, by our definition, for any in , we have that
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,
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 mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
in the variable with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
coefficients, and the ideal
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 example ...
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"
i.e.
which is the defining property of .
*Generalizing the previous example, quotient rings are often used to construct
field extensions. Suppose is some
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and is an
irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
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
with three elements. The polynomial
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 ring
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal ...
s of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
are important examples of quotient rings in
algebraic geometry. As a simple case, consider the real variety
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, 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 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
In mathematics, the system of hyperreal numbers is a way of treating Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
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
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
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
hyperinteger
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is g ...
s), see
construction of the real numbers
In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ...
.
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
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
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 In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
whenever the algebra contains a
real line 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 cla ...
.
Furthermore, the ring quotient does split into and so this ring is often viewed as the
direct sum
Nevertheless, a variation on complex numbers
is suggested by as a root of
compared to as root of
This plane of
split-complex numbers 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 radi ...
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
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the po ...
Then Hamilton’s
quaternions of 1843 can be cast as
:
If is substituted for then one obtains the ring of
split-quaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
After introduction in th ...
s. The
anti-commutative property implies that has as its square
:
Substituting minus for plus in ''both'' the quadratic binomials also results in split-quaternions.
The three types of
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
s 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, then so is ; the converse, however, is not true in general.
The natural quotient map has as its
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
; 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
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preservi ...
between the quotient ring and the image . (See also:
fundamental theorem on homomorphisms
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and ...
.)
The ideals of and are closely related: the natural quotient map provides a
bijection 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: for commutative, is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
if and only if is a
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
, while is an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
if and only if is a
prime ideal. A number of similar statements relate properties of the ideal to properties of the quotient ring .
The
Chinese remainder theorem states that, if the ideal is the intersection (or equivalently, the product) of pairwise
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
ideals , then the quotient ring is isomorphic to the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of the quotient rings , .
For algebras over a ring
An
associative algebra over a
commutative ring 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 In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:\operatorname_I R = \oplus_^\infty I^n/I^.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the gra ...
*
Residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
*
Goldie's theorem
*
Quotient module
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by ...
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 referen ...
, page 33.
* Neal H. McCoy (1948) ''Rings and Ideals'', §13 Residue class rings, page 61, Carus Mathematical Monographs #8,
Mathematical Association of America.
*
*
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 ringsfrom John Beachy's ''Abstract Algebra Online''
Ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
Ring theory