On tDimension over Strong Mori Domains

In this note we prove that if R is a strong Mori domain with t.dim R = n and with countably many prime v.ideals, then there is a chain of rings between R and R w R1=R.R2...Rn.Rw such that each R i is also a strong Mori domain and t.dim Rk = n . k + 1 for k = 1, 2, . . . , n.