THE RINGS $D((\mathcal{X}))_i$ AND $D\{\{\mathcal{X}\}\}_i$

Abstract

Let D be an integral domain, [Formula: see text] be an infinite set of indeterminates over D, and [Formula: see text] be the i th type of power series ring over D for i = 1, 2, 3. For [Formula: see text], let c(f) denote the ideal of D generated by the coefficients of f. For a star operation * on D, put [Formula: see text], where *f is the star operation of finite type on D induced by *. Let [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We show that [Formula: see text], and that D is a Noetherian domain if and only if [Formula: see text] is a Noetherian domain. We also show that D is a Krull domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain, and that D is a Dedekind domain if and only if [Formula: see text] is a Dedekind domain, if and only if [Formula: see text] is a Prüfer domain.