Rings of Formal Power Series in an Infinite Set of Indeterminates

Abstract

Let be an infinite cardinal number, be an index set of cardinality > , and X ∈ 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 ∈ over D, say, D X i for i = 1 2 3. In this paper, we let D X = ∪ D X ∈ 3 ⊆ and ≤ , and we then show that D X is an integral domain such that D X 2 D X D X 3 . We also prove that (1) D is a Krull domain if and only if D X is a Krull domain and (2) D Xis a unique factorization domain (UFD) (resp., -domain) if and only if D X 1 X n is a UFD (resp., -domain) for every integer n ≥ 1.