Universal property of the Kaplansky ideal transform and affineness of open subsets

Let R be an integral domain, I an ideal of R and ΩR(I) the Kaplansky transform of R with respect to I. A ring homomorphism α : R→A is called an I-morphism if α−1(Q)neither a superset of nor equal toI for each prime ideal Q of A. We denote by KR(I,A) the set of all the I-morphisms from R to A. It is easy to see that KR(I,−) defines a covariant functor from Ring to Set. We prove that the following statements are equivalent: (i) KR(I,−) : Ring→Set is a representable functor; (ii) the natural embedding R→ΩR(I) is an I-morphism; (iii) IΩR(I)=ΩR(I); (iv) D(I)={Pset membership, variantSpec(R) Pneither a superset of nor equal toI} is an open affine subscheme of Spec(R).