Power Series Rings over Zero-Dimensional Rings
M. Richard Sayanagi
TL;DR
This work constructs a robust non-Noetherian analogue of power series rings over 0-dimensional bases by representing the base as a directed union of large Artinian subrings and forming the ring 𝔽[[X1,...,Xn]] = ⋃α Rα[[X1,...,Xn]]. It proves faithful flatness over the base, establishes dim 𝔽[[X1,...,Xn]] = n, and shows catenarity and Noetherian-like control of prime spectra, enabling Krull's Height Theorem to hold for height-generated ideals. The results extend Cohen–Macaulay-type properties to the non-Noetherian setting via directed unions of Cohen–Macaulay subrings, yielding unmixedness and grade = height for height-generated ideals, and showing that the seven recognized non-Noetherian CM notions can all hold in the semilocal case. The construction thus provides a principled, flat, finite-dimension framework for analyzing depth-type conditions in non-Noetherian power series contexts with potential applications to non-Noetherian dimension theory and module theory.
Abstract
A power series ring over non-Noetherian rings can fail to be flat over the base ring, and its dimension can be infinite, even when the dimension of the base ring is finite. We study the case when the base ring has Krull dimension 0, and consider a version of the power series ring which preserves flatness and whose dimension remains finite. We consider properties that are well-understood in the Noetherian context, such as Krull's Height Theorem and unmixedness, and generalize them to the non-Noetherian setting.