9 - Ideals and Prime Factorization

Summary

The remaining step on the road to unique prime ideal factorization is to define "prime ideal" itself. This involves a definition of "division" for ideals. If we also define "product" of ideals, then a prime ideal is one with a property originally discovered by Euclid: if a prime p divides a product ab, then p divides a or p divides b. We then have all the ingredients needed for the definition of Dedekind domain: a Noetherian, integrally closed ring in which all prime ideals are maximal. And the main theorem follows: in a Dedekind domain, each ideal is a unique product of prime ideals.