Descent and ascent of perinormality in some ring extension settings

In this paper, we investigate some open questions regarding perinormal domains posed by Neal Epstein and Jay Shapiro in [6]. More specifically, we focus on the ascent/descent property of perinormality between “canonical” integral domain extensions, in particular, A⊂A[X] and A⊂Aˆ. We give special conditions under which perinormality ascends from A to the polynomial ring A[X] in the case of an universally catenary domain A. Whereas we have a characterizing result for when perinormality descends from A[X] to A, the sufficient condition for the descent is cumbersome to check. For this reason, we turn to special cases for which perinormality descends from A[X] to A. In the case of an analytically irreducible local domain (A,m) and its m-adic completion (Aˆ,mˆ), we refer to a technique for generating examples in which perinormality fails to ascend. When Aˆ is perinormal, we explore hypotheses under which A must be normal, perinormal, or weakly normal. Finally, we make connexions between the concepts of semi-normality, weak-normality, relative and global perinormality, and normality.