Root Closure in Commutative Rings of The Form 𝒜[[X

Abstract

Let 𝒜 = (A n ) n≥0 be an ascending chain of commutative rings with identity, and let 𝒜[[X]] be the ring of power series with coefficient of degree i in A i for each i ∈ ℕ. Thus, . In this article, we consider a ring extension 𝒜[[X]] ⊆ ℬ[[X]], where 𝒜 = (A n ) n≥0 and ℬ = (B n ) n≥0 are two chains of commutative rings such that for each i ∈ ℕ, there is a ring extension A i ⊆ B i . We give necessary and sufficient conditions for 𝒜[[X]] to be seminormal, root closed, or t-closed in ℬ[[X]].