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

✍ Scribed by Marco Fontana; Nicolae Popescu

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 151 KB
• Volume: 173
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(01)00174-8

✦ Synopsis

Let R be an integral domain, I an ideal of R and R(I ) the Kaplansky transform of R with respect to I . A ring homomorphism : R → A is called an

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; -) deÿnes 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 → R(I ) is an I -morphism; (iii) I R(I )= R(I ); (iv) D(I )={P ∈ Spec(R) | P + I } is an open a ne subscheme of Spec(R).