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Partially ordered sets of prime ideals and prime filtrations of finitely generated modules
Abstract
Let M be a finitely generated module over a Noetherian ring R. We say that M is APF-represented if it admits an associated prime filtration (APF). A natural question is: What conditions on R and M imply that M is APF-represented? In Chapter I, several rings are shown to be APF-represented as modules over themselves. For example, if R has only one associated prime ideal and that prime is principal, then R is APF-represented. Similarly, if Ass(R) consists of two principal primes P and Q with $P\notin {\rm Ass}(Q)$, then R is APF-represented. We also study the existence of APF's for submodules and quotients, and we investigate maximal APF-represented submodules if M is not APF-represented. Chapter II deals with the partially ordered sets of the prime ideals (prime spectra) of Noetherian rings. In (?) Roger Wiegand characterizes the spectrum of Z (x), and conjectures that the spectrum of every two-dimensional domain which is a finitely generated Z-algebra is order isomorphic to the spectrum of Z (x). He shows that this conjecture holds for every two-dimensional k-algebra where k is an algebraic extension of a finite field; and it holds for D (x) where D is an order in an algebraic number field. We show the conjecture holds for certain localizations and birational extensions of Z (x), e.g., Z (x, ${g\over f}\rbrack$, where (f, g) is a maximal ideal of Z (x). In Chapter III, we investigate the concept of $\Gamma$-sets in partially ordered sets. Properties of spectrum of Z (x) related to $\Gamma$-sets are discussed. We also study the projective line over the integers, Proj(Z (x)). This partially ordered set is similar to the spectrum of Z (x), but lacks one essential property: the existence of radical elements (Axiom (P5)). We give various examples where radical elements exist and where they fail to exist.
Subject Area
Mathematics
Recommended Citation
Li, Aihua, "Partially ordered sets of prime ideals and prime filtrations of finitely generated modules" (1994). ETD collection for University of Nebraska-Lincoln. AAI9516587.
https://digitalcommons.unl.edu/dissertations/AAI9516587