On Fragility of Generalizations of Factoriality

Abstract

Any class of domains, in particular a class of domains that arises from generalizations of factoriality, invites questions about its stability under the standard operations. One of these generalizations of factoriality is the one that requires that every nonzero element be contained in only finitely many principal prime ideals of height one. We use this property to settle all the open cases in the literature on stability of generalizations of factoriality under the standard ring extensions. The paper provides a compendium on the stability, under ring extensions, of all the known generalizations of factoriality. We also use stability properties of factorization in extensions of valuation domains to give a new characterization of discrete valuation domains.

Key Words:

• Factorization
• Unique factorization

2000 Mathematics Subject Classification:

• Primary 13F15

Notes

Communicated by I. Swanson.