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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