Radically finite rings and space curves

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.