Noetherian Property of Subrings of Power Series Rings

Abstract

Let R be a commutative ring with unit. We study subrings R[X; Y, λ] of R[X][[Y]] = R[X ₁,…, X _n ][[Y ₁,…, Y _m ]], where λ is a nonnegative real-valued increasing function. These rings R[X; Y, λ] are obtained from elements of R[X][[Y]] by bounding their total X-degree above by λ on their Y-degree. Such rings naturally arise from studying p-adic analytic variation of zeta functions over finite fields. Under certain conditions, Wan and Davis showed that if R is Noetherian, then so is R[X; Y, λ]. In this article, we give a necessary and sufficient condition for R[X; Y, λ] to be Noetherian when Y has more than one variable and λ grows at least as fast as linear. It turns out that the ring R[X; Y, λ] is not Noetherian for a quite large class of functions λ including functions that were asked about by Wan.

Key Words:

• Noetherian ring
• Power series ring

2010 Mathematics Subject Classification:

• 13E05
• 13F25
• 13J05

Notes

Communicated by S. Bazzoni.