Root Extensions and Factorization in Affine Domains

An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element a in R, the ascending chain of non-associate irreducible divisors in R of a(n) stabilizes on a finite set as n ranges over the positive integers, while R is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension S of R is a root extension or radical extension if for each s in S, there exists a natural number n(s) with s(n(s)) in R. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains (R, S) is governed by the relative sizes of the unit groups U(R) and U(S) and whether S is a root extension of R. The following results are deduced from these considerations: An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let R be a Noetherian domain with integral closure S. Suppose the conductor of S into R is non-zero. Then R is IDPF if and only if S is a root extension of R and U(S)/U(R) is finite.