A Characteristic Zero Non-Noetherian Factorial Ring of Dimension Three

Abstract

This paper shows the previously unknown existence of a finite dimensional non-Noetherian factorial ring in characteristic zero. The example, called “$J$", contains a field of characteristic zero and is contained in a pure transcendental extension of degree three of that field. $J$ is seen to be an ascending union of polynomial rings and degree functions are introduced on each of the polynomial rings. These are the basic facts that enable it to be seen that two extensions of $J$ are Krull. One of these extensions is a simple one and the other is a localization of $J$ at a prime ideal $P$. In the case of the latter extension, it is necessary to show that the intersection of the powers of $P$ is zero. As $J$ is the intersection of these two extensions, a theorem of Nagata is all that is needed to show then that $J$ is factorial. It is easily proved that $J$ is non-Noetherian once it is known to be factorial.