The Kaplansky Transforms Over Domains

In this paper, we characterize the Kaplansky transforms over domains. Firstly, we study the first order Kaplansky transform. We prove that if I and J are finitely generated ideals of R with R being a Priifer domain, then Ω(IJ) = Ω(I)Ω(J) = Ω(I) + Ω(J). We also show the relationship between Ω(I) and Rp with R being a Priifer domain and I being a finitely generated proper ideal of R, namely, Ω(I) (?) R_P if and only if I (?) P. And we describe the difference between the first order Kaplansky transform and the Nagata transform by an example. Besides, we also characterize the first order Kaplansky transform over Ï…-coherent domains. We prove that if R is a Ï…-coherent domain (Mori domain, SM domain), then Ω(I) is also a Ï…-coherent domain (Mori domain, SM domain). Secondly, we study the higher order Kaplansky transform and ∞-Kaplansky transform. And we obtain that if R is a Ï…-coherent domain with countably many maximal t-ideals, say P₁, P₂, … , P_s, …, then t-dim Ω^∞ = t-dim R — 1. Finally, we describe the applications of the Kaplansky transforms. We characterize the Ω_B-ideals. And we indicate that there is the only Ω_B-ideal J such that Ω(I, B) = Ω(J, B) for each nonzero ideal I of R. Besides, we also describe the quasi Ω-domains and the ω-integraHy closed quasi Ω-domains. Moreover, we prove that if R is a PVMD and if R is a quasi Ω-domain, then for each nonzero prime ω-ideal P of R, either ω-(?){ω-(?){P)) = w-P↓ or ω-(?)(ω-(?)(P)) = ω-P↓\{P} with P unbranched.