The Ohio University-Ohio State University ring theory seminar series presents Byung Gyun Kang discussing “Krull dimension of power series rings” on Friday, Jan. 25, at 4:45 p.m. in Cockins Hall 240, OSU-Columbus.
Gyun Kang is Professor of Mathematics at Pohang University of Science and Technology, South Korea.
Abstract: We prove that the Krull dimension of the power series ring over a nonSFT domain is at least aleph_2. In particular, the Krull dimension of the power series ring over the ring of algebraic integers is aleph_2 and the height of P[[x]] is aleph_2 as well for each nonzero prime ideal P of the ring of algebraic integers under the Continuum Hypothesis. A ring D is called an SFT ring if for each ideal I of D, there exists a finitely generated ideal J of D with J \subseteq I and a positive integer k such that a^k \in J for all a \in I. A ring is nonSFT if it is not SFT.
For a cardinal number a and a ring D, we say that dim(D) \geq a if D has a chain of prime ideals with length \geq a. J. T. Arnold showed that if D is a non-SFT ring then dim(D[[x]]) \geq alpeh_0. Let C be the class of non-SFT domains. The class C includes the class of finite-dimensional non-discrete valuation domains, the class of non-Noetherian almost Dedekind domains, the class of completely integrally closed domains that are not Krull domains, the class of integral domains with non-Noetherian prime spectrum, and the class of integral domains with a nonzero proper idempotent ideal. The ring of algebraic integers , the ring of integer-valued polynomials on Z and the ring of entire functions are also members of the class C. In this talk we prove that dim(D[[x]]) \geq 2^aleph_1 for every D \in C and that under the continuum hypothesis 2^aleph_1 is the greatest lower bound of dim(D[[x]]) for D \in C. On the other hand there exists a (finite dimensional) SFT domain D such that dim(D[[x]]) \geq 2^aleph_1.
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