On S-GCD domains

Abstract

Let [Formula: see text] be a multiplicative set in an integral domain [Formula: see text]. A nonzero ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-[Formula: see text]-principal if there exist an [Formula: see text] and [Formula: see text] such that [Formula: see text]. Call [Formula: see text] an [Formula: see text]-GCD domain if each finitely generated nonzero ideal of [Formula: see text] is [Formula: see text]-[Formula: see text]-principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and [Formula: see text]-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017) 2869–2879]. One aim of this paper is to characterize [Formula: see text]-GCD domains, giving several equivalent conditions and showing that if [Formula: see text] is an [Formula: see text]-GCD domain then [Formula: see text] is a GCD domain but not conversely. Also we prove that if [Formula: see text] is an [Formula: see text]-GCD [Formula: see text]-Noetherian domain such that every prime [Formula: see text]-ideal disjoint from [Formula: see text] is a [Formula: see text]-ideal, then [Formula: see text] is [Formula: see text]-factorial and we give an example of an [Formula: see text]-GCD [Formula: see text]-Noetherian domain which is not [Formula: see text]-factorial. We also consider polynomial and power series extensions of [Formula: see text]-GCD domains. We call [Formula: see text] a sublocally [Formula: see text]-GCD domain if [Formula: see text] is a [Formula: see text]-GCD domain for every non-unit [Formula: see text] and show, among other things, that a non-quasilocal sublocally [Formula: see text]-GCD domain is a generalized GCD domain (i.e. for all [Formula: see text] is invertible).