Well-centered overrings of an integral domain

✍ Scribed by William Heinzer; Moshe Roitman

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 276 KB
• Volume: 272
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00462-9

✦ Synopsis

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B ⊆ K of A that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A. If B = A[b] is a simple extension of A, we prove that B is a localization of A if and only if B is flat and well-centered over A. If the integral closure of A is a Krull domain, in particular, if A is Noetherian, we prove that every finitely generated flat wellcentered overring of A is a localization of A. We present examples of (non-finitely generated) flat well-centered overrings of a Dedekind domain that are not localizations.