Perinormality in Polynomial and Module-Finite Ring Extensions
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In this dissertation we investigate some open questions posed by Epstein and Shapiro in  regarding perinormal domains. More specifically, we focus on the ascent/descent property of perinormality between "canonical" integral domain extensions, in particular, R ? R[X] and R ? Rb. We give special conditions under which perinormality ascends from R to the polynomial ring R[X] in the case that R is a universally catenary domain. Whereas we have a characterizing result for when perinormality descends from R[X] to R, the sufficient condition for the descent is cumbersome to check. For this reason, we turn to special cases for which perinormality descends from R[X] to R. In the case of an analytically irreducible local domain (R, m) and its m-adic completion (R, b mRb), we refer to a technique for generating examples in which perinormality fails to ascend. When Rb is perinormal, we explore hypotheses under which R must be normal, perinormal, or weakly normal.