Weak Normality and Strong t-closedness of Generalized Power Series Rings

Abstract

Abstract. For an extension A ⊆ B of commutative rings, we present a sufficient condi-tion for the ring [[A S, ≤ ]] of generalized power series to be weakly normal (resp., stronglyt-closed) in [[B S, ≤ ]], where (S, ≤) be a torsion-free cancellative strictly ordered monoid.As a corollary, it can be applied to the ring of power series in infinitely many indetermi-nates as well as in finite indeterminates. 1. Introduction and preliminariesLet A ⊆ B be an extension of commutative rings with (the same) identity.Consider the following conditions:(a) B is integral over A.(b) Spec(B) → Spec(A) is a bijection.(c) The residue field extensions are isomorphisms. i.e., for each Q ∈ Spec(B) theextension A P /PA P ,→ B Q /QB Q is an isomorphism, where P = Q∩A.(c 0 ) The residue field extensions are purely inseparable.We first recall some special extensions satisfying two or three conditions aboveincluding the condition (a).• R. G. Swan called the extension A ⊆ B subintegral if (a), (b) and (c) aresatisfied.