Two counterexamples in Power Series Rings

Abstract

In Seminormality in Power Series Rings, Brewer and Nichols ask whether R rcd seminormal implies R[[ X]] red is seminormal. They prove that the implication holds if R is Noetherian ( J. Algebra 92 (1983), 282–284). The purpose of this paper is to show that if R is not Noetherian the implication need not hold. In other terminology, R Pic regular does not imply R[[ X]] Pic regular. In the discussion of whether the example had the desired (or undesired) properties, the following question arose. If ∑ ∝ i=0 a i X i is nilpotent in R[[ X]], must the degree of nilpotency of the { a i } in R be bounded? That the a i , are nilpotent is known. The paper will give an example where (∑ a i X i ) 2 = 0 but there is no N so a N i = 0, for all i.