On a theorem by Brewer

One of the most frequently referenced monographs on power series rings, “Power Series over Commutative Rings” by James W. Brewer, states in Theorem 21 that if M is a non-SFT maximal ideal of a commutative ring R with identity, then there exists an infinite ascending chain of prime ideals in the power series ring $R?X?$, $Q_0?Q_1???Q_n??$ such that $Q_n\cap R=M$ for each n. Moreover, the height of $M?X?$ is infinite. In this paper, we show that the above theorem is false by presenting two counter examples. The first counter example shows that the height of $M?X?$ can be zero (and hence there is no chain $Q_0?Q_1???Q_n??$ of prime ideals in $R?X?$ satisfying $Q_n\cap R=M$ 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 infinite, there may be no infinite chain $\{Q_n\}$ of prime ideals in $R?X?$ satisfying $Q_n\cap R=M$ for each n.