On a Theorem by Brewe

Abstract

One of the most frequently referenced monographs on power series rings, “Power Series over Commutative Rings” by James W. Brewer, states in Theo- rem 21 that if M is a non-SFT maximal ideal of a commutative ring R with identity, then there exists an in?nite ascending chain of prime ideals in the power series ring R[[X]], Q0 도 Q1 도 ··· 도 Qn 도 ··· such that Qn ∩ R = M for each n. Moreover, the height of M [[X ] is in?nite. In this thesis, we show that the above theorem is false by presenting two counter examples. The ?rst counter example shows that the height of M [[X ] can be zero (and hence there is no chain Q0 도 Q1 도 ··· 도 Qn 도 ··· of prime ideals in R[[X ] satisfying Qn ∩ R = M for each n). In this example, the ring R is one-dimensional. In the second counter example, we prove that even if the height of M [[X ] is uncountably in?nite, there can be no in?nite chain {Qn} of prime ideals in R[[X ] satisfying Qn ∩ R = M for each n. In each of the two counter examples, we completely describe the spectrum of the corresponding ring, determine the SFT property of P and calculate (or give possibilities for) height of P [[X ] for each prime ideal P of the ring.