When $(D[[X]])\sb{P[[X]]}$ is a valuation ring

Abstract

Let D be an integral domain with identity and let K denote the quotient field of D. If P is a prime ideal of D denote by P[[X]] that prime ideal of D[[X]] consisting of all those formal power series each of whose coefficients belongs to P. In this paper the following question is considered: When is (D[[X]h)r[[x]] a valuation ring? Our main theorem states that if (D[[X]])p[[x]] is a valuation ring, then Dp must be a rank one discrete valuation ring. Moreover, we show that if Dp is a rank one discrete valuation ring and if PD[[X]]=P[[X]], then (D[[X]])p[[x]] is a valuation ring. We also give an example to show that (D[[X]])Prrx]] need not be a valuation ring when Dp is rank one discrete. 1. Main theorem. Let V be a rank one valuation ring with quotient field K and let M denote the maximal ideal of V. If v is a valuation on K associated with V, then we may take the value group of v to be a subgroup of the additive group of real numbers. Forf= >j0afXi, define v*(f)= inf0o,.:,{v(ft)}. Equipped with this terminology, we have LEMMA 1. v* is a valuation on V[[X]]. Moreover, MV[[X]] is a prime ideal of V[[X]] and (V[[X]])_jsv[[x]] is the valuation ring associated with v*. PROOF. Letf= Z;tOfiX?, g = T!o giXi E V[ [X]]. It is straightforward to verify that v*(f+g)>min{v*(f), v*(g)}. To show that v*(fg)= v*(f)+v*(g), it suffices to show that v*(fg)?v*(f)+v*(g). We first consider the case in which there exist coefficients fr and g, of f and g respectively, with the property that v*(f)=v(fr) and v*(g)=v(g,). If r and s are minimal with this property, then v ( fig) = v*(f) + v*(g) i+i=r+s and since (Zi+i=r+sfig) is the coefficient of Xr+s in fg, it follows that v*(fg)?v*(f)+v*(g). In particular, if v is a discrete valuation, this shows that v* is a valuation. Assuming now that v is a nondiscrete valuation, let f, g be arbitrary elements of V[[X]] and let m E M\(O). Iff '= :E?_ i i Presented to the Society, November 20, 1970 under the title Essential valuation overrings of D[[X]]; received by the editors February 29, 1972. AMS (MOS) subject classiflcations (1969). Primary 1393; Secondary 1398.