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

[摘要] Let R be an integral domain, I an ideal of R and Omega(R)(I) the Kaplansky transform of R with respect to I. A ring homomorphism alpha: R --> A is called an I-morphism if alpha(-1) (Q) not superset of or equal to I for each prime ideal Q of A. We denote by K-R(I,A) the set of all the I-morphisms from R to A. It is easy to see that K-R(I,-) defines a covariant functor from Ring to Set. We prove that the following statements are equivalent: (i) K-R(I,-):Ring --> Set is a representable functor; (ii) the natural embedding R --> Omega(R)(I) is an I-morphism; (iii) IOmega(R)(I)=Omega(R)(I); (iv) D(I)={P is an element of Spec(R) \ P not superset of or equal to I} is an open affine subscheme of Spec(R). (C) 2002 Elsevier Science B.V. All rights reserved.

[发布日期] 2002-08-24