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Locally factorial integral domains
Journal article   Open access   Peer reviewed

Locally factorial integral domains

D. D Anderson and David F Anderson
Journal of algebra, Vol.90(1), pp.265-283
1984
DOI: 10.1016/0021-8693(84)90214-X
url
https://doi.org/10.1016/0021-8693(84)90214-XView
Published (Version of record) Open Access

Abstract

Following Fossum (9, p. 8 11, we define an integral domain R to be locally factorial if R, = R [ l/f] is factorial for each nonzero nonunit f E R. For example, any factorial integral domain or one-dimensional quasilocal integral domain is locally factorial. We show that a locally factorial integral domain R which is not quasilocal is a Krull domain, and R, is factorial for each maximal ideal M of R. We will be mainly interested in locally factorial Krull domains. In this case we can relate divisibility properties of R to group-theoretic properties of Cl(R), its divisor class group. The first examples of Dedekind domains which are locally factorial (PID), but not factorial (PID), are due to Claborn [6]. We know of no locally factorial Krull domains which are neither factorial nor Dedekind domains. Most of our results will be stated for arbitrary locally factorial Krull domains and subintersections of Krull domains. However, major emphasis will be towards Dedekind domains. In this case a factorial overring is just a PID, and each overring is a subintersection. In Section 2, we prove several basic results about locally factorial integral domains and give several examples of locally factorial Krull domains. Claborn’s examples actually satisfy the stronger property that each proper overring of R is factorial (PID). In Section 3, we study those Krull domains each of whose proper overrings is factorial. Such an integral domain which is not quasilocal is necessarily a Dedekind domain with cyclic divisor class 265

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