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
$\mathcal{O}$
be the ring of algebraic integers. Under the continuum hypothesis, we show that
$\mathcal{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$ contains an
$\left\mathcal{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]\right$generated
(and hence uncountably generated) ideal which is not a
$\beta $generated ideal for any
cardinal number
$\beta <\left\mathcal{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]\right$ and that
the concepts “${\aleph}_{1}$generated
ring” and “${\aleph}_{0}$generated
ring” are different for the power series ring
$R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$.
