Classes of rings torsion-free over their centers

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.