A remark on the noetherian property of power series rings

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 f 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[[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 alpha, it is proved that R is an alpha-generated ring if and only if R[[X]] 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[[X]] is a Noetherian ring. Let O be the ring of algebraic integers. Under the continuum hypothesis, we show that O[[X]] contains an vertical bar O[[X]]vertical bar-generated (and hence uncountably generated) ideal which is not a beta-generated ideal for any cardinal number beta < vertical bar O[[ X]]vertical bar j and that the concepts "aleph(1)-generated ring" and "aleph(0)-generated ring" are different for the power series ring R[[X]].