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

Abstract

Let Clt(A) denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping Clt(A)→Clt(A〚X〛), is injective and if A is a regular UFD, then Clt(A)→Clt(A〚X〛), is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping Clt(A)→Clt(A〚X〛); [I]↦[(I.A〚X〛)t] is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for Clt(A)→Clt(A〚X〛), to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an s∈S and a∈I such that sI⊆aA⊆I. The S-class group of A, S-Clt(A) is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-Clt(A)≃S-Clt(A〚X〛).