For a positive integer n, the ring Z/nZ is a domain if and only if n is prime. This notion of prime will be generalized to arbitrary ideals. Later, the notion of residue classes will be extended beyond Z/nZ and Q[X]/(d) to residue class rings with respect to arbitrary ideals, and it will turn out that primality has the same role as for Z/nZ.
Let R be a commutative ring and let I be an ideal of
R. We say that I is proper if I
R.
Definition
- I is called a prime ideal if it is proper and, for all a,
b
R,
ab I implies a I or b I. - I is called maximal if it is proper and if there exists no proper ideal strictly containing I.
Although the definitions look very different, there are important connections between the two notions. For instance one implies the other.
Theorem
A maximal ideal is prime.