Quotient ring explained

R and a two-sided ideal I in R , a new ring, the quotient ring R / I , is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations. (Quotient ring notation always uses a fraction slash "/".)

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 R , we may define an equivalence relation \sim on R as follows:a \sim b if and only if a - b is in I .Using the ideal properties, it is not difficult to check that \sim is a congruence relation.In case a \sim b , 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 2 Z , the even integers). The equivalence class of the element a in R is given by:$$\backslash left[a\; \backslash right]\; =\; a\; +\; I\; :=\; \backslash left\backslash lbrace\; a\; +\; r\; :\; r\; \backslash in\; I\; \backslash right\backslash rbrace$$

This 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

R / I ; it becomes a ring, the factor ring or quotient ring of R modulo I , if one defines(a + I) + (b + I) = (a + b) + I ;(a + I)(b + I) = (ab) + I .(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of R / I is \bar{ 0 } = ( 0 + I) = I , and the multiplicative identity is \bar{ 1 } = ( 1 + I) .

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 or the canonical homomorphism.

Examples

• The quotient ring

R / \lbrace 0 \rbrace is naturally isomorphic to R , and R / R is the zero ring \lbrace 0 \rbrace , since, by our definition, for any r \in R , we have that \left[ r \right] = r + R = \left\lbrace r + b : b \in R \right\rbrace , which equals R itself. This fits with the rule of thumb that the larger the ideal I , the smaller the quotient ring R / I . If I is a proper ideal of R , i.e., I ≠ R , then R / I is not the zero ring.

• Consider the ring of integers

Z and the ideal of even numbers, denoted by 2 Z . Then the quotient ring Z / 2 Z has only two elements, the coset 0 + 2 Z consisting of the even numbers and the coset 1 + 2 Z consisting of the odd numbers; applying the definition, \left[ z \right] = z + 2 Z = \left\lbrace z + 2 y : 2 y \in 2 Z \right\rbrace , where 2 Z is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, F _{2} . 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 is essentially arithmetic in the quotient ring Z / n Z (which has n elements).

• Now consider the ring of polynomials in the variable

X with real coefficients, R [X] , and the ideal I = \left( X ^{2} + 1 \right) consisting of all multiples of the polynomial X ^{2} + 1 . The quotient ring R [X] / ( X ^{2} + 1 ) is naturally isomorphic to the field of complex numbers C , with the class [X] playing the role of the imaginary unit i . The reason is that we "forced" X ^{2} + 1 = 0 , i.e. X ^{2} = -1 , which is the defining property of i . Since any integer exponent of i must be either \pm i or \pm 1 , that means all possible polynomials essentially simplify to the form a + bi . (To clarify, the quotient ring R [X] / ( X ^{2} + 1 ) is actually naturally isomorphic to the field of all linear polynomials aX + b; a,b \in R , where the operations are performed modulo X ^{2} + 1 . In return, we have X ^{2} = -1 , 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 K[X] . Then L = K[X] / (f) is a field whose minimal polynomial over K is f , which contains K as well as an element x = X + (f) .

• One important instance of the previous example is the construction of the finite fields. Consider for instance the field

F _{3} = Z / 3 Z 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 F _{3} [X]/(f) . This is a field with 3 ^{2} = 9 elements, denoted by F _{9} . 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 R ^{2} . The ring of real-valued polynomial functions defined on V can be identified with the quotient ring R [X,Y] / (X ^{2} - Y ^{3)} , and this is the coordinate ring of V . The variety V is now investigated by studying its coordinate ring.

• Suppose

M is a C ^{infty} -manifold, and p is a point of M . Consider the ring R = C ^{infty} (M) of all C ^{infty} -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 f ). Then the quotient ring R / I is the ring of germs of C ^{infty} -functions on M at p .

• Consider the ring

F of finite elements of a hyperreal field ^{*} R . 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 ^{*} R , together with 0 , is an ideal in F , and the quotient ring F / I is isomorphic to the real numbers R . The isomorphism is induced by associating to every element x of F the standard part of x , i.e. the unique real number that differs from x by an infinitesimal. In fact, one obtains the same result, namely R , 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

R [X] / (X) , R [X] / (X + 1 ) , and R [X] / (X - 1 ) are all isomorphic to R and gain little interest at first. But note that R [X] / (X ^{2)} 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 X ^{2} . This variation of a complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent.

Furthermore, the ring quotient

R [X] / (X ^{2} - 1 ) does split into R [X] / (X + 1 ) and R [X] / (X - 1 ) , so this ring is often viewed as the direct sum R ⊕ R .Nevertheless, a variation on complex numbers z = x + yj is suggested by j as a root of X ^{2} - 1 = 0 , compared to i as root of X ^{2} + 1 = 0 . This plane of split-complex numbers normalizes the direct sum R ⊕ R 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 R \langle X, Y \rangle . Then Hamilton's quaternions of 1843 can be cast as:$$\backslash mathbb\; \backslash langle\; X,\; Y\; \backslash rangle\; /\; (X^2\; +\; 1,\backslash ,\; Y^2\; +\; 1,\backslash ,\; XY\; +\; YX)$$

If

Y ^{2} - 1 is substituted for Y ^{2} + 1 , 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

R \langle X, Y, Z \rangle and constructing appropriate ideals.

Properties

Clearly, if

R is a commutative ring, then so is R / I ; 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 I . More precisely, given a two-sided ideal I in R and a ring homomorphism f: R \to S whose kernel contains I , 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([a]) = f(a) for all a in R . 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 im (f) . (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 I , and we write M / I for the corresponding ideal in R / I (i.e. M / I = p(M) ), the quotient rings R / I and (R / I) / (M / I) are naturally isomorphic via the (well-defined) mapping a + M \mapsto (a + I) + M / I .

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

R ≠ \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 R / I .

The Chinese remainder theorem states that, if the ideal

I is the intersection (or equivalently, the product) of pairwise coprime ideals I _{1,} \ldots, I _{k} , then the quotient ring R / I is isomorphic to the product of the quotient rings R / I _{n,} n = 1 , \ldots, k .

For algebras over a ring

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

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

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