Valuations on power series rings in an arbitrary set of indeterminates

Let [Formula: see text] be an integral domain and [Formula: see text] be a nonempty (either finite or infinite) set of indeterminates over [Formula: see text]. There are three types of power series rings in the set [Formula: see text] over [Formula: see text], denoted by [Formula: see text], [Formula: see text], respectively. In general, [Formula: see text] and the two containments can be strict. In this paper, we show that if [Formula: see text] is a rank one valuation domain with valuation [Formula: see text], then [Formula: see text] defined by [Formula: see text] is a valuation on [Formula: see text]. Moreover, [Formula: see text] is a prime ideal of [Formula: see text] and [Formula: see text] is the valuation domain associated with [Formula: see text]. We also show that, for [Formula: see text], if [Formula: see text] is a valuation domain, then both [Formula: see text] and [Formula: see text] are DVRs. Our results generalize those by Arnold and Brewer in which [Formula: see text] has a single indeterminate. In proving the results, we show that for each [Formula: see text], [Formula: see text] is completely integrally closed if and only if [Formula: see text] is completely integrally closed.