Dedekind Domains Sample Clauses

Dedekind Domains. In order to extend the concept of an integer, some provisions must be made. It is a well known fact that the unique factorization of the integers does not necessarily hold in finite degree extensions of the rational numbers. This is due specifically to the fact that the concepts of prime and irreducible are equivalent under certain circumstances, but they are not equivalent in general. Fortunately, Dedekind domains provide a different kind of unique factorization if we are willing to replace the idea of a prime number with that of a prime ideal. We assume the reader knows the definitions of the most basic terms: ring, subring, and ideal. But, we will take some time to review the following terms in order to stress completeness. Also, we assume that our rings are commutative and contain a multiplicative identity denoted as 1. The following material comes from the references [5] and [21].