Projectively full radical ideals in integral extension rings

Let R be a Noetherian commutative ring with unit 1≠0, and let I be a regular proper ideal of R. The main question considered in this paper is whether there exists a finite integral extension ring A of R for which the nilradical of IA is a projectively full ideal that is projectively equivalent to IA. A related and stronger question that we also consider is whether there exists a finite integral extension ring A of R for which the nilradical J of IA is projectively equivalent to IA and for which all the Rees integers of J are one. The following two results are special cases of the main theorems in the present paper: (1) If R is a Noetherian integral domain, then there exists a finite integral extension ring A of R such that the nilradical of IA is projectively equivalent to IA. (2) If also R contains a field of characteristic zero, then there exists a finite free integral extension ring A of R for which the nilradical of IA is a projectively full ideal that is projectively equivalent to IA.