Valuations on power series rings in an arbitrary set of indeterminates

Let D be an integral domain and X = {X_λ}_{λ ∈ Λ} be a nonempty (either finite or infinite) set of indeterminates over D. There are three types of power series rings in the set X over D, denoted by D[[X]]_i, i = 1, 2, 3, respectively. In general, D[[X]]₁ ⊆ D[[X]]₂ ⊆ D[[X]]₃ and the two containments can be strict. In this paper, we show that if D = V is a rank one valuation domain with valuation v, then v^∗ - defined by v^∗ (f) = inf{v(a) | a is a coefficient of f} is a valuation on V [[X]]_i,i = 1, 2, 3. Moreover, MV [[X]]_i is a prime ideal of V [[X]]_i and (V [[X]]_i)_{MV [[X]]i} is the valuation domain associated with v^∗. We also show that, for i = 1, 2, 3, if (D[[X]]_i)_P[[X]]i is a valuation domain, then both D_P and (D[[X]]_i)_P[[X]]i are DVRs. Our results generalize those by Arnold and Brewer in which has a single indeterminate. In proving the results, we show that for each i = 1, 2, 3, D[[X]]_i is completely integrally closed if and only if D is completely integrally closed.