  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.
