A New Characterization of Principal Ideal Domains

Abstract

In 2008 N. Q. Chinh and P. H. Nam characterized principal ideal domains as integral domains that satisfy the following two conditions: (i) they are unique factorization domains, and (ii) all maximal ideals in them are principal. We improve their result by giving a characterization in which each of these two conditions is weakened. At the same time we improve a theorem by P. M. Cohn which characterizes principal ideal domains as atomic Bézout domains. We will also show that every PC domain is AP and that the notion of PC domains is incomparable with the notion of pre-Schreier domains (hence with the notions of Schreier and GCD domains as well).2010 Mathematics Subject Classification. Primary 13F15; Secondary 13A05, 13F10.

For which additive submonoids $M$ of nonnegative rationals is $F[X;M]$ AP?

“…It is not AP, otherwise it would be a UFD, but it is not, as X • XY 2 = XY • XY are two decompositions of X 2 Y 2 into non-associated atoms. The PC domains were introduced in our paper [6], where all the implications involving this type of domains can be found.…”

Factorization in integral domains.

“…Note: Every k-generated ideal, k ≥ 1, is an l-generated ideal for all l > k. Of specific interest to this chapter are those domains whose every 2−generated ideal is principal; we call these integral domains Bézout domains. Our next theorem, presented in our paper [5], improves both of the above theorems. It weakens one of the conditions in Cohn's theorem and both conditions in Chinh and Nam's theorem.…”