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]].

Key Words:

• Power series rings
• Root closed
• Seminormal
• t-Closed

2000 Mathematics Subject Classification:

• 13F25
• 13F45
• 13Bxx

ACKNOWLEDGMENT

The author would like to thank the referee for his/her careful considerations.

Notes

Communicated by I. Swanson.