A remark on the Noetherian property of power series rings

Abstract

Let α be a (finite or infinite) cardinal number. An ideal of a ring R is called an α-generated ideal if it can be generated by a set with cardinality at most α. A ring R is called an α-generated ring if every ideal of R is an α-generated ideal. When α is finite, the class of α-generated rings has been studied in literature by scholars such as I. S. Cohen and R. Gilmer. In this paper, the class of α-generated rings when α is infinite (in particular, when α = ℵ 0 , the smallest infinite cardinal number) is considered. Surprisingly, it is proved that the concepts "ℵ 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 α, it is proved that R is an α-generated ring if and only if R[X] is an α-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 ᏻ be the ring of algebraic integers. Under the continuum hypothesis, we show that ᏻ [[X]] contains an |ᏻ[[X]]|-generated (and hence uncountably generated) ideal which is not a β-generated ideal for any cardinal number β < |ᏻ [[X]]| and that the concepts "ℵ 1 -generated ring" and "ℵ 0 -generated ring" are different for the power series ring R [[X]].

Some strange behaviors of the power series ring R[[X]]

“…We let J be the ideal of R [[X]] generated by all f s with s ∈ A. The following is the main result of [10]. Proof.…”

“…Even though the class of ℵ 0 -generated rings is strictly larger than the class of Noetherian rings, it is shown in [10] that when restricted to power series rings, they are actually the same. In other words, the concepts "ℵ 0 -generated ring" and "Noetherian ring" are the same for the power series ring R [[X]].…”

“…[10, Theorem 22]) For any infinite cardinal number α, a ring R is an αgenerated ring if and only if the polynomial ring R[X] is an α-generated ring.ITM Web of Conferences 20, 01002 (2018) https://doi.org/10.1051/itmconf/20182001002 ICM 2018…”