Doctoral Dissertations

Date of Award

5-2000

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

David E. Dobbs

Committee Members

David Anderson, Robert McConnel, Mark Kot

Abstract

This work develops criteria for determining whether an extension of rings R ⊆ T satisfies the condition that A ⊆ B satisfies P for each ring extension such that R ⊆ A ⊆ B ⊆ T (i.e.,(R,T)is a "P-pair"), for some properties P related to"going-down."In Chapter I, we define many properties and associated pairs of rings related to "going-down" and study how these ring-theoretic properties behave under the formation of factor domains,finite products, and reduced rings.In Chapter II, assuming the (Krull) dimension of the base ring R is zero, we find that (R,T) is a "going-down-ring pair" (resp., "open-ring pair") if and only if,for each minimal prime P of T, we have that the transcendence degree of >i>T/P over R/(P ∩ R) is less than or equal to one (resp., is zero). We provide examples to show that transcendence degree criteria are not sufficent to characterize going-down-ringpairs or open-ring pairs in the case where the dimension of the base ring is nonzero,and then we provide some positive partial results involving INC-pairs. In Chapter III, we generalize results of Dobbs and Papick on extensions of going-down domains and open domains and we use these generalizations to show that a going-down-ring pair (resp., open-ring pair)is a going-down-pair (resp., open-pair) if the top ring has a unique minimal prime ideal.In Chapter IV,using results of Dobbs, Mulay,and Akiba, we show that if the base ring R is Prüfer or the top ring T is a field, then a pair of domains (R,T)is a flat pair if and only if if R ⊆ T is either an algebraic field extension or the inclusion of a Prüfer domain inside an overring. We also prove some results on i-pairs of arbitrary rings.

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