On the class group and S-class group of formal power series rings

On S-GCD domains

“…Definition 2.1. [14] Let D be an integral domain and S a multiplicative subset of D. We say that a nonzero ideal I of D is S-υ-principal if there exist an s ∈ S and a ∈ D such that sI ⊆ aD ⊆ I υ . We also define D to be an S-GCD domain if each finitely generated nonzero ideal of D is S-υ-principal.…”

“…In a GCD domain, every finite type υ-ideal of D is principal, thus a GCD domain is a PVMD. This property can be generalized in several different ways ( [2], [14]). However, we will be mostly interested in the S-GCD property ( [14]).…”

“…This property can be generalized in several different ways ( [2], [14]). However, we will be mostly interested in the S-GCD property ( [14]). Let S be a multiplicative subset of D and I a nonzero ideal of D. We say that I is S-principal (resp., S-υ-principal ) if there are s ∈ S and a ∈ I (resp., a ∈ I υ ) such that sI ⊆ aD.…”

“…Let S be a multiplicative subset of D and I a nonzero ideal of D. We say that I is S-principal (resp., S-υ-principal ) if there are s ∈ S and a ∈ I (resp., a ∈ I υ ) such that sI ⊆ aD. Following [14], D is an S-GCD domain if each finitely generated nonzero ideal of D is S-υ-principal. Note that if S consists of units of D, then D is an S-GCD domain if and only if it is a GCD domain.…”

“…We give with an additional condition a necessary and sufficient condition for the power series ring D[[X]] to be an S-GCD domain. First, recall from [14], the power series ring D[[X]] is said to satisfy the property ( * ), if for all integral υ-invertible υ-ideals I and J of D In the third section of this paper, we define the notion of a sublocally s-GCD domain. Let D be an integral domain and s a nonzero element of D. We say that D is an s-GCD domain if for each pair a, b ∈ D * , there is a positive integer n and an element c ∈ aD ∩ bD such that s n (aD ∩ bD) ⊆ cD.…”