Going-down pairs of communicative rings

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.