König's theorem (set theory)

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra. One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I, essentially by requiring that all elements of I be zero in R. Intuitively, the quotient ring R/I is a "simplified version" of R where the elements of I are "ignored".

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 ~ on R as follows:

a ~ b if and only if a − b is in I.

Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by

[a] = a + I := { a + r : r in I }.

This equivalence class is also sometimes written as a mod 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) = (a b) + I.

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of R/I is (0 + I) = I, and the multiplicative identity is (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 R/{0} is naturally isomorphic to R, and R/R is the zero ring {0}. This fits with the general 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 2Z. Then the quotient ring Z/2Z has only two elements, zero for the even numbers and one for the odd numbers. It is naturally isomorphic to the finite field with two elements, F₂. 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/nZ (which has n elements).

• Now consider the ring R[X] of polynomials in the variable X with real coefficients, and the ideal I = (X² + 1) consisting of all multiples of the polynomial X² + 1. The quotient ring R[X]/(X² + 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: we "forced" X² + 1 = 0, i.e. X² = −1, which is the defining property of i.

• 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 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 F₃[X]/(f). This is a field with 3²=9 elements, denoted by F₉. 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 = {(x,y) | x² = y³ } as a subset of the real plane R². The ring of real-valued polynomial functions defined on V can be identified with the quotient ring R[X,Y]/(X² − Y³), and this is the coordinate ring of V. The variety V is now investigated by studying its coordinate ring.

• Suppose M is a C^∞-manifold, and p is a point of M. Consider the ring R = C^∞(M) of all C^∞-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^∞-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.

Alternative 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]/(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 X². This alternative complex plane 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 R[X]/(X − 1), so this ring is often viewed as the direct sum R $\oplus$ R. Nevertheless, an alternative complex number z = x + y j is suggested by j as a root of X² − 1, compared to i as root of X² + 1 = 0. This plane of split-complex numbers normalizes the direct sum $R\oplus R$ by providing a basis {1, j } 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 alternatives

Hamilton’s quaternions of 1843 can be cast as R[X,Y]/(X² + 1, Y² + 1, XY + YX). If Y² − 1 is substituted for Y² + 1, then one obtains the ring of split-quaternions. Substituting minus for plus in both the quadratic binomials also results in split-quaternions. The anti-commutative property YX = −XY implies that XY has for its square

(XY)(XY) = X(YX)Y = −X(XY)Y = − XXYY = −1.

The three types of biquaternions can also be written as quotients by conscripting the three-indeterminate ring R[X,Y,Z] 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 → S whose kernel contains I, then there exists precisely one ring homomorphism g : R/I → 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 → 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/M and (R/I)/(M/I) are naturally isomorphic via the (well-defined!) mapping a + M ↦ (a+I) + M/I.

In commutative algebra and algebraic geometry, the following statement is often used: If R ≠ {0} is a commutative ring and I is a maximal ideal, then the quotient ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain. 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 I₁,...,I_k, then the quotient ring R/I is isomorphic to the product of the quotient rings R/I_p, p=1,...,k.

See also

• Residue field
• Goldie's theorem

Notes

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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.
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• 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

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