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.

Key Words:

• Krull domain
• Power series ring in an uncountable set of indeterminates
• UFD
• π-domain

2010 Mathematics Subject Classification:

• 13J05
• 13F05
• 13A15

ACKNOWLEDGMENTS

The author would like to thank the referee for several valuable comments and suggestions.

Notes

Communicated by S. Bazzoni.