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