Rings, Fields and Their First Properties

Here a ring has a multiplicative identity 1. Addition forms an abelian group: sums commute and associate, 0 is an identity, and additive inverses exist. Multiplication is closed and associative, with 1a = a = a1. Both distributive laws hold: a(b + c) = ab + ac and (a + b)c = ac + bc. Multiplication need not commute. Subtraction is always available as addition of an additive inverse.

An integral domain is a commutative ring with 1 not equal to 0 and no nonzero factors whose product is zero. A field is a commutative ring with 1 not equal to 0 in which every nonzero element has a multiplicative inverse. Rationals form a field. Integers form a domain, but division by an integer need not stay integral. Even integers lack a multiplicative identity and are not a ring under our convention.

A nonzero element a is a zero divisor when ab = 0 for some nonzero b. In a commutative ring, cancellation of a nonzero factor works exactly when there are no zero divisors: ab = ac gives a(b - c) = 0. In a field, multiply ab = 0 by a inverse when a is nonzero, obtaining b = 0. Thus every field is an integral domain. Zero itself is excluded from our zero-divisor definition.

For a positive integer n, Z_n has residues 0 through n - 1, with sums and products reduced modulo n. These operations form a commutative ring. A residue a is invertible exactly when gcd(a, n) = 1. Bezout's identity then gives integers s and t with sa + tn = 1; reducing modulo n gives sa = 1, so s is an inverse. Conversely, an inverse equation shows every common divisor of a and n divides 1.

For prime p, each residue 1 through p - 1 has gcd 1 with p, so Bezout gives every nonzero residue an inverse. Thus Z_p is a field. If n is composite, write n = ab with 1 < a, b < n. The two nonzero residues a and b have product 0, so Z_n is not a field. Together with the Z1 case, Z_n is a field exactly when the positive modulus n is prime.

Polynomials Q[x] form a commutative ring with identity. The product of two nonzero polynomials has degree equal to the sum of their degrees, so it is nonzero. Therefore Q[x] is a domain. It is not a field: multiplying the polynomial x by any nonzero polynomial gives positive degree, never the constant 1. Constants such as nonzero rational numbers still have inverses.