On a theorem by Brewer |
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Authors: | Le Thi Ngoc Giau Byung Gyun Kang |
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Affiliation: | Department of Mathematics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea |
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Abstract: | 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 , such that for each n. Moreover, the height of 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 can be zero (and hence there is no chain of prime ideals in satisfying 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 is uncountably infinite, there may be no infinite chain of prime ideals in satisfying for each n. |
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Keywords: | 13A15 13C15 13F25 |
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