Quotient ring

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In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, 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. Intuitively, the quotient ring R/I is a "simplified version" of R where the elements of I are "ignored".

Quotient rings are unrelated to the quotient field, or field of fractions, of an integral domain, and also unrelated to the rings of quotients resulting from localization of rings.

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 ba 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) = (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 (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 most extreme examples of quotient rings are provided by modding out the most extreme ideals, {0} and R itself. R/{0} is naturally isomorphic to R, and R/R is the trivial ring {0}. This fits with the general rule of thumb that the smaller the ideal I, the larger the quotient ring R/I. If I is a proper ideal of R, i.e. IR, then R/I won't be the trivial 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, one 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 which contains K as well as an element x = X + (f) whose minimal polynomial over K is 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. 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.

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 : RS whose kernel contains I, then there exists precisely one ring homomorphism g : R/IS 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 : RS 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,...,Ik, then the quotient ring R/I is isomorphic to the product of the quotient rings R/Ii.

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