Bases of ideals and Rees valuation rings

Let I be a regular proper ideal in a Noetherian ring R. We prove that there exists a simple free integral extension ring A of R such that the ideal IA has a Rees-good basis; that is, a basis c1,…,cg such that ciW=IW for i=1,…,g and for all Rees valuation rings W of IA. Moreover, A may be constructed so that: (i) IA and I have the same Rees integers (with possibly different cardinalities), and (ii) AP is unramified over RP∩R for each asymptotic prime divisor P of IA. Indeed, if H is a regular ideal in R such that each asymptotic prime divisor of H is contained in an asymptotic prime divisor of I, then (ii) holds for HA. If Card(ReesH)⩽Card(ReesI), we prove that (i) also holds for HA and H. If I=(b1,…,bg)R and b1,…,bg is an asymptotic sequence, we prove that b1,…,bg is a Rees-good basis of I.