Radically finite rings and space curves

TL;DR

This paper introduces the concept of radically finite rings, explores their properties, and characterizes when polynomial rings over certain domains are radically finite, revealing implications for space curves and set-theoretic complete intersections.

Contribution

It defines radically finite rings, characterizes their properties, and establishes conditions under which polynomial rings over specific domains are radically finite, linking to geometric properties of space curves.

Findings

Finite dimensional radically finite rings are Noetherian.

Polynomial rings over certain domains are radically finite iff the base domain is a Dedekind domain with torsion class group.

Not all space curves are set-theoretic complete intersections.

Abstract

We define radically finite rings and show that finite dimensional radically finite rings are Noetherian, and that if either R is a finite character Hilbert domain that contains a field of characteristic zero or a finite dimensional Prufer domain, then the polynomial ring R[X] over R is radically finite if and only if R is a Dedekind domain with torsion ideal class group. We then consider the radically finite condition on UFD and show that there does not exist a finite character UFD R of Krull dimension 2 over which the polynomial ring R[X] is radically finite. From this it follows that not all space curves are set theoretic complete intersection.