Quotient ring explained

in, a new ring, the quotient ring, is constructed, whose elements are the cosets of

subject to special

and

⋅

operations. Quotient ring notation almost always uses a fraction slash ""; stacking the ring over the ideal using a horizontal line as a separator is uncommon and generally avoided.

Quotient rings are distinct from the "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

\sim

as follows:

a\simb

if and only if

a-b

is in .Using the ideal properties, it is not difficult to check that

\sim

is a congruence relation.In case, we say that

and

are congruent modulo

(for example,

and

are congruent modulo

as their difference is an element of the ideal, the even integers). The equivalence class of the element

is given by:

\left[a \right] = \overline = a + I := \left\lbrace a + r : r \in I \right\rbrace

This equivalence class is also sometimes written as

a\bmodI

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

R / I

is, and the multiplicative identity is .

The map

from

R / I

defined by

p(a)=a+I

is a surjective ring homomorphism, sometimes called the natural quotient map, natural projection map, or the canonical homomorphism.

Examples

The quotient ring

R / \lbrace0\rbrace

is naturally isomorphic to, and

R/R

is the zero ring, since, by our definition, for any, 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

R/I

is not the zero ring.

Consider the ring of integers

and the ideal of even numbers, denoted by . Then the quotient ring

Z/2Z

has only two elements, the coset

0+2Z

consisting of the even numbers and the coset

1+2Z

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, then every integer is either

(if it is even) or

(if it is odd and therefore differs from an even number by). Modular arithmetic is essentially arithmetic in the quotient ring

Z/nZ

(which has

elements).

Now consider the ring of polynomials in the variable

with real coefficients,, and the ideal

I=\left(X²+1\right)

consisting of all multiples of the polynomial . The quotient ring

R[X] / (X²+1)

is naturally isomorphic to the field of complex numbers, with the class

[X]

playing the role of the imaginary unit . The reason is that we "forced", i.e., which is the defining property of . Since any integer exponent of

must be either

\pmi

or, that means all possible polynomials essentially simplify to the form . (To clarify, the quotient ring is actually naturally isomorphic to the field of all linear polynomials, where the operations are performed modulo . In return, we have, and this is matching

to the imaginary unit in the isomorphic field of complex numbers.)

Generalizing the previous example, quotient rings are often used to construct field extensions. Suppose

is some field and

is an irreducible polynomial in . Then

L=K[X] / (f)

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

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 . This is a field with

3²=9

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. As a simple case, consider the real variety

V=\left\lbrace(x,y)|x²=y³\right\rbrace

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

C^infty

-manifold, and

is a point of . Consider the ring

R=C^infty(M)

of all

C^infty

-functions defined on

and let

be the ideal in

consisting of those functions

which are identically zero in some neighborhood

(where

may depend on). Then the quotient ring

R / I

is the ring of germs of

C^infty

-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

-n<x<n

exists. The set

of all infinitesimal numbers in, together with, is an ideal in, and the quotient ring

F / I

is isomorphic to the real numbers . The isomorphism is induced by associating to every element

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.

Real quadratic algebras

See main article: Quadratic algebra. The quotients,, and

R[X]/(X-1)

are all isomorphic to

and gain little interest at first. But note that

R[X]/(X²⁾

is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of

R[X]

by . This quadratic algebra arises as a subalgebra whenever the algebra contains a real line and a nilpotent.

Furthermore, the ring quotient

R[X]/(X²-1)

does split into

R[X]/(X+1)

and, so this algebra is often viewed as the direct sum .Nevertheless, a quadratic algebra

z=x+yj

is defined by

as an element of the ideal of, compared to

as root of . This plane of split-complex numbers normalizes the direct sum

R ⊕ R

by providing a basis

\left\lbrace1,j\right\rbrace

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. The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry.

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:

\mathbb \langle X, Y \rangle / (X^2 + 1,\, Y^2 + 1,\, XY + YX)

Y²-1

is substituted for, then one obtains the ring of split-quaternions. The anti-commutative property

YX=-XY

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

R\langleX,Y,Z\rangle

and constructing appropriate ideals.

Properties

Clearly, if

is a commutative ring, then so is ; the converse, however, is not necessarily true.

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

R / I

are essentially the same as the ring homomorphisms defined on

that vanish (i.e. are zero) on . More precisely, given a two-sided ideal

and a ring homomorphism

f:R\toS

whose kernel contains, there exists precisely one ring homomorphism

g:R / I\toS

with

gp=f

(where

is the natural quotient map). The map

here is given by the well-defined rule

g([a])=f(a)

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

f:R\toS

induces a ring isomorphism between the quotient ring

R / \ker(f)

and the image . (See also: Fundamental theorem on homomorphisms.)

The ideals of

and

R / I

are closely related: the natural quotient map provides a bijection between the two-sided ideals of

that contain

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

is a two-sided ideal in

that contains, and we write

M / I

for the corresponding ideal in

R / I

(i.e.), the quotient rings

R / M

and

(R/I) / (M/I)

are naturally isomorphic via the (well-defined) mapping .

The following facts prove useful in commutative algebra and algebraic geometry: for

R ≠ \lbrace0\rbrace

commutative,

R / I

is a field if and only if

is a maximal ideal, while

R/I

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

R / I

is isomorphic to the product of the quotient rings .

For algebras over a ring

over a commutative ring

is itself a ring. If

is an ideal in

(closed under

-multiplication:), then

A/I

inherits the structure of an algebra over

and is the quotient algebra.

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.
Book: Joseph Rotman . Galois Theory. Springer . 21–23 . 1998 . 0-387-98541-7 . 2nd .
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", pp. 47–51.

External links

- Ideals and factor rings from John Beachy's Abstract Algebra Online

Notes and References

Book: Nathan Jacobson . Jacobson . Nathan . Structure of Rings . American Mathematical Soc. . 1984 . revised . 0-821-87470-5.
Book: Dummit . David S. . Foote . Richard M. . Abstract Algebra . . 2004 . 3rd . 0-471-43334-9.
Book: Lang, Serge . Serge Lang . Algebra . . . 2002 . 0-387-95385-X.