#### Vol. 283, No. 2, 2016

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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 $\alpha$ 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 $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$. In other words, if every ideal of $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$ is countably generated, then each of them is in fact finitely generated. This shows a strange behavior of the power series ring $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$ 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 $\mathsc{O}$ be the ring of algebraic integers. Under the continuum hypothesis, we show that $\mathsc{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$ contains an $|\mathsc{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]|$-generated (and hence uncountably generated) ideal which is not a $\beta$-generated ideal for any cardinal number $\beta <|\mathsc{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]|$ and that the concepts “${\aleph }_{1}$-generated ring” and “${\aleph }_{0}$-generated ring” are different for the power series ring $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$.

##### Keywords
countably generated ideal, $n$-generated ideal, Noetherian ring, polynomial ring, power series ring
##### Mathematical Subject Classification 2010
Primary: 03E10, 13A15, 13B25, 13E05, 13F25