Rings of Formal Power Series in an Infinite Set of Indeterminates

Abstract

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X λ}λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X λ}λ∈Λ over D, say, D[[{X λ}]] i for i = 1, 2, 3. In this paper, we let D[[{X λ}]]α = ∪ {D[[{X λ}λ∈Γ]]3 | Γ ⊆ Λ and |Γ| ≤ α}, and we then show that D[[{X λ}]]α is an integral domain such that D[[{X λ}]]2 ⊊ D[[{X λ}]]α ⊊ D[[{X λ}]]3. We also prove that (1) D is a Krull domain if and only if D[[{X λ}]]α is a Krull domain and (2) D[[{X λ}]]α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X 1,…, X n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.