ABSTRACT
21.1 There are not many commandments in mathematics, but there is certainly one: “You shall not divide by zero!” On the other hand it is allowed to have divisors of zero. The set of zero divisors is not an ideal of the ring, so one cannot get rid of it by a simple construction of some quotient ring. This makes restricting attention to rings without zero divisors (called : domains) into a real restriction. The kernel of a ring homomorphism to a domain is always a prime ideal, the kernel of a ring homomorphism to a field is a maximal ideal. Domains are related to fields, e.g. a domain is embeddable into a field, and so there are several interesting relations between prime and maximal ideals. In this section we begin to develope some basic ideal theory. Observe that finite (possibly noncommutative) domains are division rings (Proposition 21.23), and we do encounter some finite fields like ℤp that appears as the prime field in any domain of characteristic p ≠ 0.
