Let J() denote the intersection
of the maximals ideals of a ring. The following properties are studied, for a ring R
torsion-free over its center C: (i) J(R) ∩ C = J(C); (ii) “Going up” from
prime ideals of C to prime ideals of R; (iii) If M is a maximal ideal of R
then M ∩ C is a maximal ideal of C; (iv) if M is a maximal (resp. prime)
ideal of C, then M = MR ∩ C. Properties (i)–(iv) are known to hold for
many classes of rings, including rings integral over their centers or finite
modules over their centers. However, using an idea of Cauchon, we show
that each of (i)–(iv) has a counterexample in the class of prime Noetherian
PI-rings.
