A remark on the Noetherian property of power series rings

Byung Gyun Kang and Phan Thanh Toan

Vol. 283 (2016), No. 2, 353–363

DOI: 10.2140/pjm.2016.283.353

Abstract

Let $\alpha$ be a (finite or infinite) cardinal number. An ideal of a ring $R$ is called an $\alpha$-generated ideal if it can be generated by a set with cardinality at most $\alpha$. A ring $R$ is called an $\alpha$-generated ring if every ideal of $R$ is an $\alpha$-generated ideal. When $\alpha$ is finite, the class of $\alpha$-generated rings has been studied in literature by scholars such as I. S. Cohen and R. Gilmer. In this paper, the class of $\alpha$-generated rings when $\alpha$ is infinite (in particular, when $\alpha ={\aleph }_0$, the smallest infinite cardinal number) is considered. Surprisingly, it is proved that the concepts “${\aleph }_0$-generated ring” and “Noetherian ring” are the same for the power series ring $R\left[\left[X\right]\right]$. In other words, if every ideal of $R\left[\left[X\right]\right]$ is countably generated, then each of them is in fact finitely generated. This shows a strange behavior of the power series ring $R\left[\left[X\right]\right]$ compared to that of the polynomial ring $R\left[X\right]$. Indeed, for any infinite cardinal number $\alpha$, it is proved that $R$ is an $\alpha$-generated ring if and only if $R\left[X\right]$ is an $\alpha$-generated ring, which is an analogue of the Hilbert basis theorem stating that $R$ is a Noetherian ring if and only if $R\left[X\right]$ is a Noetherian ring. Let $\mathcal{O}$ be the ring of algebraic integers. Under the continuum hypothesis, we show that $\mathcal{O}\left[\left[X\right]\right]$ contains an $\left|\mathcal{O}\left[\left[X\right]\right]\right|$-generated (and hence uncountably generated) ideal which is not a $\beta$-generated ideal for any cardinal number $\beta <\left|\mathcal{O}\left[\left[X\right]\right]\right|$ and that the concepts “${\aleph }_1$-generated ring” and “${\aleph }_0$-generated ring” are different for the power series ring $R\left[\left[X\right]\right]$.

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