A generalized approximation theorem for Dedekind domains

Abstract

It is well known that a Dedekind wvith a finite number of prime ideals is a principal ideal domain. reasonable generalization of this result would be: If is a Dedekind and S is the set of prime ideals of A, then card S<card implies that is a principal ideal domain. In fact, this latter statement is false; see [1]. But it is true that if (card S)o<card A, then is a principal ideal domain. proof is given in the present article of a slight generalization (analogous to the weak approximation theorem) of this result. Before proceeding to this result, we give a proposition that displays a large class of examples for which the stronger assertion of the first paragraph is valid. We will use the phrase Let A, S be a Dedekind domain rather than Let be a Dedekind domain, and let S be the set of prime ideals of A for the balance of the article. Also, if P is a prime ideal of A, then v, will denote the normed valuation going with the prime ideal P.