1. Descent and ascent of perinormality in some ring extension settings
- Author
-
Andrew McCrady and Dana Weston
- Subjects
Pure mathematics ,Ring (mathematics) ,Algebra and Number Theory ,Polynomial ring ,media_common.quotation_subject ,010102 general mathematics ,Extension (predicate logic) ,01 natural sciences ,Integral domain ,0103 physical sciences ,Domain (ring theory) ,010307 mathematical physics ,0101 mathematics ,Focus (optics) ,Normality ,Mathematics ,Descent (mathematics) ,media_common - Abstract
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 ⊂ A ˆ . 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 ( A ˆ , m ˆ ) , we refer to a technique for generating examples in which perinormality fails to ascend. When A ˆ 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.
- Published
- 2018