In
ring theory, a branch of
abstract algebra, 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 and to the
quotient space in
linear algebra.
It is a specific example of a
quotient, as viewed from the general setting of
universal algebra. Starting with a
ring and a
two-sided ideal in , a new ring, the quotient ring , is constructed, whose elements are the
cosets 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 on as follows:
:
if and only if 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. Compare
coset and
quotient group.) The zero-element of is
and the multiplicative identity is
The map from to defined by
is a
surjective ring homomorphism, sometimes called the ''natural quotient map'' or the ''
canonical homomorphism''.
Examples
*The quotient ring is
naturally isomorphic to , and is the
zero ring 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
integers 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 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
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
is essentially arithmetic in the quotient ring (which has elements).
*Now consider the
ring of polynomials in the variable with
real coefficients, and the ideal
consisting of all multiples of the
polynomial The quotient ring is naturally isomorphic to the field of
complex numbers with the class playing the role of the
imaginary unit . 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 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
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 rings of
algebraic varieties are important examples of quotient rings in
algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...
. 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 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 and a
nilpotent.
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 may be compared to the
unit circle of the
ordinary complex plane.
Quaternions and variations
Suppose and are two, non-commuting,
indeterminates and form the
free algebra Then Hamilton’s
quaternions of 1843 can be cast as
:
If is substituted for then one obtains the ring of
split-quaternions. 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
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, 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 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 and
algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...
: 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. 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 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 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
*
Quotient module
Notes
Further references
* F. Kasch (1978) ''Moduln und Ringe'', translated by DAR Wallace (1982) ''Modules and Rings'',
Academic Press, 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 theory