The Theory Of Pre-Krull Domains

In this paper, we characterize pre-Krull domains by using general star operations. Firstly, we study the relationship between pre-Krull domains and several other important domains. We prove that R is a pre-Krull domain of finite character satisfying ascending chain conditions over principal ideals of the localization of R if and only if R is a Krull domain. We also show that each overring of R is a pre-Krull domain , then R is a generalized Dedekind domain and Prufer domain. And we indicate that each t-linked overring of a pre-Krull domain is also a pre-Krull domain if the quotient ring of the polynomial domain over a pre-Krull domain is a pre-Krull domain. Moreover, we also indicate that the localization of a pre-Krull domain at a prime v-ideal is a discrete valuation domain. Besides, we prove that the integral closure R of R is a Priifer domain with P^-1 ≠ R[X] for any fractional ideal P of UTZ(R), then R is a UMV domain. Secondly, we study the polynomial domains and w-dimension about pre-Krull domains and UMT domains. We prove that R is a pre-Krull domain if and only if i?[{X_α}] is a pre-Krull domain, if and only if R[{X_α}]_Nv is a generalized Dedekind domain, and if and only if R[{X_α}]_Nv is a pseudo-principal ideal domain . Moreover, we show that R is a pre-Krull domain satisfying the condition of (P), then R[[X]]_Nv is a pre-Krull domain. We also obtain that R is a UMT domain, then w-dimR=w-dim(R[{X_α}]). Finally, we characterize UMT domains, PVMDs and pre-Krull domains in group rings. We show that R is a UMT domain, then R[X; G] is a UMT domain. And we prove that R is a PVMD if and only if R[X; G]is a PVMD , if and only if R[X; G]_Nv is a PVMD . Moreover, R is a pre-Krull domain if and only if R[X;G]_Nv is a pre-Krull domain. We also obtain that R is a UMT domain, then w-dimR=w-dimR[X; G].