Rings of very strong finite type

TL;DR

This paper introduces the VSFT property, a natural variant of the SFT condition, and investigates its fundamental properties and relationship with the original SFT condition in the context of commutative rings.

Contribution

It defines and studies the VSFT property, providing insights into its fundamental properties and how it compares to the existing SFT condition.

Findings

VSFT is a natural variant of SFT.

VSFT ideals and rings have distinct properties.

Comparison shows differences and similarities with SFT.

Abstract

The SFT (for strong finite type) condition was introduced by J. Arnold in the context of studying the condition for formal power series rings to have finite Krull dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub the VSFT (for very strong finite type) property. We explore some of the fundamental properties of VSFT ideals and rings and compare and contrast with the known SFT condition.