On the generalized Krull property in power series rings

Abstract

One open problem in commutative algebra and field arithmetic posed by Jarden is whether the power series ring R[X] is a generalized Krull domain if R is a generalized Krull domain. Assuming R is a generalized Krull domain, Paran and Temkin proved that R[X] is a generalized Krull domain if and only if R[X] is a Krull domain. Hence, if R is a generalized Krull domain that is not a Krull domain, then R[X] is never a generalized Krull domain. In this paper, we show that the assumption R is a generalized Krull domain in Paran and Temkin's result can be dropped. In other words, R[X] is a generalized Krull domain if and only if R[X] is a Krull domain and hence if and only if R is a Krull domain. (C) 2020 Elsevier B.V. All rights reserved.