Almost Splitting Sets and AGCD Domains#

Abstract

Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 ≠ d ∈ D, there exists an n = n(d) with d n = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that x n D ∩ y n D is a principal ideal. We prove that the polynomial ring D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ⊆ D′[X] is a root extension, where D′ is the integral closure of D. We also show that D + XD S [X] is an AGCD domain if and only if D and D S [X] are AGCD domains and S is an almost splitting set.