On the Ascent of Accp to Simple Overrings

ABSTRACT

In this article, we raise the question as to what conditions permit a simple overring of a domain R —that is, a domain of the form for some f, g  ∈  R such that g ≠ 0—to inherit the ascending chain condition on principal ideals from R . Our main theorem reveals that, if g is a prime element of R , the complete answer can be found by considering the Archimedean property. We then, in turn, use this theorem to establish equivalent conditions for a certain class of simple overring to inherit the property of being a unique factorization domain from R .

Keywords:

• Archimedean domain
• Ascending chain condition on principal ideals
• Birational extension
• Noetherian domain
• Simple overring
• Unique factorization domain

2000 Mathematics Subject Classification:

• Primary 13B99
• 13E15
• 13F15
• Secondary 13A15
• 13E05
• 13G05

Notes

^aIt should be noted that Li bases, Li (2000b, Proposition 2) on Li (2000a, Proposition 2.3) in which an error appears in the proof. However, Rush corrects this error in his review of Li (2000a) (see MR1739356 (2001c:13027)).

^#Communicated by I. Swanson.