Vol. 283, No. 2, 2016

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ISSN: 0030-8730
A remark on the Noetherian property of power series rings

Byung Gyun Kang and Phan Thanh Toan

Vol. 283 (2016), No. 2, 353–363
Abstract

Let α be a (finite or infinite) cardinal number. An ideal of a ring R is called an α-generated ideal if it can be generated by a set with cardinality at most α. A ring R is called an α-generated ring if every ideal of R is an α-generated ideal. When α is finite, the class of α-generated rings has been studied in literature by scholars such as I. S. Cohen and R. Gilmer. In this paper, the class of α-generated rings when α is infinite (in particular, when α = 0, the smallest infinite cardinal number) is considered. Surprisingly, it is proved that the concepts “0-generated ring” and “Noetherian ring” are the same for the power series ring R[[X]]. In other words, if every ideal of R[[X]] is countably generated, then each of them is in fact finitely generated. This shows a strange behavior of the power series ring R[[X]] compared to that of the polynomial ring R[X]. Indeed, for any infinite cardinal number α, it is proved that R is an α-generated ring if and only if R[X] is an α-generated ring, which is an analogue of the Hilbert basis theorem stating that R is a Noetherian ring if and only if R[X] is a Noetherian ring. Let O be the ring of algebraic integers. Under the continuum hypothesis, we show that O[[X]] contains an |O[[X]]|-generated (and hence uncountably generated) ideal which is not a β-generated ideal for any cardinal number β < |O[[X]]| and that the concepts “1-generated ring” and “0-generated ring” are different for the power series ring R[[X]].

Keywords
countably generated ideal, $n$-generated ideal, Noetherian ring, polynomial ring, power series ring
Mathematical Subject Classification 2010
Primary: 03E10, 13A15, 13B25, 13E05, 13F25
Milestones
Received: 29 November 2015
Revised: 15 January 2016
Accepted: 17 January 2016
Published: 22 June 2016
Authors
Byung Gyun Kang
Department of Mathematics
Pohang University of Science and Technology
Pohang 37673
South Korea
Phan Thanh Toan
Division of Computational Mathematics and Engineering
Institute for Computational Science
Ton Duc Thang University
Ho Chi Minh City
Vietnam
Faculty of Mathematics and Statistics
Ton Duc Thang University
Ho Chi Minh City
Vietnam