Quotient ring

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 $R$ and a two-sided ideal $I$ in , a new ring, the quotient ring , is constructed, whose elements are the cosets of $I$ in $R$ subject to special $+$ and $\cdot$ 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 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 $R$ and a two-sided ideal $I$ in , we may define an equivalence relation $\sim$ on $R$ as follows:
: $a \sim b$ 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 $a$ and $b$ are ''congruent modulo'' $I$ (for example, $1$ and $3$ are congruent modulo $2$ as their difference is an element of the ideal , the even integers). The equivalence class of the element $a$ in $R$ is given by:
\left a \right= \overline = a + I := \left\lbrace a + r : r \in I \right\rbraceThis equivalence class is also sometimes written as $a \bmod I$ and called the "residue class of $a$ modulo $I$".

The set of all such equivalence classes is denoted by ; it becomes a ring, the factor ring or quotient ring of $R$ 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 $p$ from $R$ to $R\ /\ I$ 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 $R\ /\ \lbrace 0 \rbrace$ is naturally isomorphic to , and $R / R$ is the zero ring , since, by our definition, for any , we have that , which equals $R$ itself. This fits with the rule of thumb that the larger the ideal , the smaller the quotient ring . If $I$ is a proper ideal of , i.e., , then $R / I$ is not the zero ring.
* Consider the ring of integers $\mathbb$ and the ideal of even numbers, denoted by . Then the quotient ring $\mathbb / 2 \mathbb$ has only two elements, the coset $0 + 2 \mathbb$ consisting of the even numbers and the coset $1 + 2 \mathbb$ consisting of the odd numbers; applying the definition, , where $2 \mathbb$ 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 $0$ (if it is even) or $1$ (if it is odd and therefore differs from an even number by ). Modular arithmetic is essentially arithmetic in the quotient ring $\mathbb / n \mathbb$ (which has $n$ elements).
* Now consider the ring of polynomials in the variable $X$ with real coefficients, , and the ideal $I = \left( X^2 + 1 \right)$ consisting of all multiples of the polynomial . The quotient ring \mathbb /\ ( X^2 + 1 ) is naturally isomorphic to the field of complex numbers , with the class /math> 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 $i$ must be either $\pm i$ 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 $X$ 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 $K$ is some field and $f$ is an irreducible polynomial in . Then L = K /\ (f) is a field whose minimal polynomial over $K$ is , which contains $K$ as well as an element .
* One important instance of the previous example is the construction of the finite fields. Consider for instance the field $F_3 = \mathbb / 3\mathbb$ with three elements. The polynomial $f(X) = Xi^2 +1$ is irreducible over $F_3$ (since it has no root), and we can construct the quotient ring . This is a field with $3^2 = 9$ 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 $V = \left\lbrace (x,y) , x^2 = y^3 \right\rbrace$ as a subset of the real plane . The ring of real-valued polynomial functions defined on $V$ can be identified with the quotient ring , and this is the coordinate ring of . The variety $V$ is now investigated by studying its coordinate ring.
* Suppose $M$ is a $\mathbb^$-manifold, and $p$ is a point of . Consider the ring $R = \mathbb^(M)$ of all $\mathbb^$-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 ). Then the quotient ring $R\ /\ I$ is the ring of germs of $\mathbb^$-functions on $M$ at .
* Consider the ring $F$ 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 $x$ for which a standard integer $n$ with $-n < x < n$ exists. The set $I$ 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 $x$ of $F$ the standard part of , i.e. the unique real number that differs from $x$ by an infinitesimal. In fact, one obtains the same result, namely , if one starts with the ring $F$ of finite hyperrationals (i.e. ratio of a pair of hyperintegers), see construction of the real numbers.

Variations of complex planes

The quotients , , and \mathbb / (X - 1) are all isomorphic to $\mathbb$ and gain little interest at first. But note that \mathbb / (X^2) is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of \mathbb /math> 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 \mathbb / (X^2 - 1) does split into \mathbb / (X + 1) and , so this ring is often viewed as the direct sum .
Nevertheless, a variation on complex numbers $z = x + yj$ is suggested by $j$ as a root of , compared to $i$ as root of . This plane of split-complex numbers normalizes the direct sum $\mathbb \oplus \mathbb$ by providing a basis $\left\lbrace 1, j \right\rbrace$ 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.

Quaternions and variations

Suppose $X$ and $Y$ are two non-commuting indeterminates and form the free algebra . Then Hamilton's quaternions of 1843 can be cast as:
$\mathbb \langle X, Y \rangle / ( X^2 + 1,\, Y^2 + 1,\, XY + YX )$

If $Y^2 - 1$ is substituted for , then one obtains the ring of split-quaternions. The anti-commutative property $YX = -XY$ implies that $XY$ 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 $\mathbb \langle X, Y, Z \rangle$ and constructing appropriate ideals.

Properties

Clearly, if $R$ is a commutative ring, then so is ; the converse, however, is not true in general.

The natural quotient map $p$ has $I$ 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 $R$ that vanish (i.e. are zero) on . More precisely, given a two-sided ideal $I$ in $R$ and a ring homomorphism $f : R \to S$ whose kernel contains , there exists precisely one ring homomorphism $g : R\ /\ I \to S$ with $gp = f$ (where $p$ is the natural quotient map). The map $g$ here is given by the well-defined rule g( = f(a) for all $a$ 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 $f : R \to S$ induces a ring isomorphism between the quotient ring $R\ /\ \ker (f)$ and the image . (See also: '' Fundamental theorem on homomorphisms''.)

The ideals of $R$ and $R\ /\ I$ are closely related: the natural quotient map provides a bijection 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 , 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 $R \neq \lbrace 0 \rbrace$ commutative, $R\ /\ I$ is a field if and only if $I$ is a maximal ideal, while $R / I$ is an integral domain if and only if $I$ is a prime ideal. A number of similar statements relate properties of the ideal $I$ to properties of the quotient ring .

The Chinese remainder theorem states that, if the ideal $I$ 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

An associative algebra $A$ over a commutative ring $R$ is a ring itself. If $I$ is an ideal in $A$ (closed under $R$-multiplication), then $A / I$ inherits the structure of an algebra over $R$ 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", pp. 47–51.

External links

* {{springer, title=Quotient ring, id=p/q076920

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

Ring
Ring theory