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 ``", contains a field of characteristic zero and is contained in a pure transcendental extension of degree three of that field. 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 are Krull. One of these extensions is a simple one and the other is a localization of at a prime ideal . In the case of the latter extension, it is necessary to show that the intersection of the powers of is zero. As is the intersection of these two extensions, a theorem of Nagata is all that is needed to show then that is factorial. It is easily proved that is non-Noetherian once it is known to be factorial.
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Keywords: Factorial, Krull dimension, Noetherian, polynomial ring, degree function
Article copyright: © Copyright 1973 American Mathematical Society