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.