文摘
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 A0; with identity, then there exists an infinite ascending chain of prime ideals in the power series ring a050646c45e341f4" title="Click to view the MathML source">R〚X〛, e601c6b157957629b804a1063e4f7" title="Click to view the MathML source">Q0⊊Q1⊊⋯⊊Qn⊊⋯ such that 98c614973fb6fcf092d3aa78" title="Click to view the MathML source">Qn∩R=M for each n . Moreover, the height of ab8fd5db7afe94c878f0369" title="Click to view the MathML source">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 ab8fd5db7afe94c878f0369" title="Click to view the MathML source">M〚X〛 can be zero (and hence there is no chain e601c6b157957629b804a1063e4f7" title="Click to view the MathML source">Q0⊊Q1⊊⋯⊊Qn⊊⋯ of prime ideals in a050646c45e341f4" title="Click to view the MathML source">R〚X〛 satisfying 98c614973fb6fcf092d3aa78" title="Click to view the MathML source">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 ab8fd5db7afe94c878f0369" title="Click to view the MathML source">M〚X〛 is uncountably infinite, there may be no infinite chain {Qn} of prime ideals in a050646c45e341f4" title="Click to view the MathML source">R〚X〛 satisfying 98c614973fb6fcf092d3aa78" title="Click to view the MathML source">Qn∩R=M for each n.