Characterizations Of Krull Type Domains

Abstract:In this paper, we characterize Krull type domains byusing general star operations. Firstly, we study the properties of Krull typedomains. We prove that R is a Krull type domain if and only if R[x] is a Krulltype domain, if and only if R[X]Nv is also a Krull type domain, if and onlyif R[X]Nv is a Pru¨fer domain and R[X]Nv has finite character, if and only ifR[X]Nv is a B′ezout domain and R[X]Nv has finite character. Moreover, wealso show that each finite generated faith ?at ideal of a Krull type domain isa projective ideal. Besides, we indicate that each nonzero ideal A of a Krulltype domain is a strongly w-type 2 ideal. Secondly, we study the relationshipbetween Krull type domains and several other important domains. We provethat R is a valuation domain if and only if R is a local Krull type domain andR is a TL domain and Spec(R) is totally ordered. Moreover, We have showedthat R is a Krull domain if and only if R is a Krull type domain and R is a Hdomain and dim(R)=1. Besides, we obtain the equivalent conditions betweenKrull type domains and GCD domains. And we indicate that R is a Krulltype domain, then R[[X]] is not necessary to be a Krull type domain. We alsoprove that a Krull type domain R is a Pru¨fer domain, then R is a Archimedeandomain if and only if b is contained in some minimal prime ideal of R for everynonzero nonunit b. Finally, we study the extensions of Krull type domains. Weshow that each t-linked overring of a Krull type domain is a Krull type domain.Moreover, we prove that the w-integral closure RLw of a Krull type domain hasPIT property if and only if w-dim(R) = 1. We also obtain that a Krull typedomain R is a TL domain and Pseudo-SM domain, then Rwg = K.