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• W3043152631 endingPage "322" @default.
• W3043152631 startingPage "306" @default.
• W3043152631 abstract "An ideal I of a commutative ring R with identity is called an SFT (strong finite type) ideal if there exist a finitely generated ideal J of R with J⊆I and a positive integer k such that ak∈J for each a∈I. A ring R is called an SFT ring if every ideal of R is an SFT ideal. Arnold showed that if R is a non-SFT ring, then the Krull dimension of the power series ring R〚X〛 is infinite. Let C be the class of non-SFT rings. In this paper, we show that dim⁡R〚X〛≥2ℵ1 for every R∈C. In other words, if R is a non-SFT ring, then there exists a chain of prime ideals in R〚X〛 with length at least 2ℵ1. We also prove that under the continuum hypothesis 2ℵ1 is the greatest lower bound of dim⁡R〚X〛 for R∈C. If M is a non-SFT maximal ideal of a ring R such that M is the radical of a countably generated ideal, then we construct a chain {Pα} of prime ideals in R〚X〛 with length at least 2ℵ1 lying between MR〚X〛 and M〚X〛, i.e., MR〚X〛⊆Pα⊆M〚X〛 for each α. The same result holds when M is the non-SFT maximal ideal of a zero-dimensional quasi-local ring or a one-dimensional quasi-local domain R." @default.
• W3043152631 created "2020-07-23" @default.
• W3043152631 creator A5061755555 @default.
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• W3043152631 date "2020-11-01" @default.
• W3043152631 modified "2023-10-14" @default.
• W3043152631 title "Krull dimension of power series rings" @default.
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• W3043152631 doi "https://doi.org/10.1016/j.jalgebra.2020.06.028" @default.
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