3 - Prime Ideals

Summary

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab ∈ P, it follows that a ∈ P or b ∈ P; equivalently, P is a prime ideal if and only if the factor ring R/P is a domain. (The terminology comes from algebraic number theory, where, for instance, one replaces the prime numbers in ℤ by the prime ideals in a Dedekind domain in order to preserve the unique factorization property.) The importance of prime ideals is perhaps clearest in the setting of algebraic geometry, for if R is the coordinate ring of an affine algebraic variety, the prime ideals of R correspond to irreducible subvarieties.

In the noncommutative setting, we define an integral domain just as we do in the commutative case (as a nonzero ring in which the product of any two nonzero elements is nonzero), but it turns out not to be a good idea to concentrate our attention on ideals P such that R/P is a domain. In fact, many noncommutative rings have no factor rings which are domains.