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〛R〚X〛, Q0⊊Q1⊊⋯⊊Qn⊊⋯Q0⊊Q1⊊⋯⊊Qn⊊⋯ such that Qn∩R=MQn∩R=M for each n  . Moreover, the height of M〚X〛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〛M〚X〛 can be zero (and hence there is no chain Q0⊊Q1⊊⋯⊊Qn⊊⋯Q0⊊Q1⊊⋯⊊Qn⊊⋯ of prime ideals in R〚X〛R〚X〛 satisfying Qn∩R=MQn∩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〛M〚X〛 is uncountably infinite, there may be no infinite chain {Qn}{Qn} of prime ideals in R〚X〛R〚X〛 satisfying Qn∩R=MQn∩R=M for each n.