Quotient Ring - Examples

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. I ≠ R, 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, 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 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/(X2 + 1) is naturally isomorphic to the field of complex numbers C, with the class 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. Then L = K/(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/(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/(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.