Recently, N. Epstein and J. Shapiro introduced and studied the perinormal domains: those domains $A$ whose overrings satisfying going down over $A$ are flat $A$-modules. We show that every Prufer v-multiplication domain is perinormal and has no proper lying over overrings. Conversely, we show that a w-treed perinormal domain is a Prufer v-multiplication domain. We give two pull-back constructions that produce perinormal/non-perinormal domains.
"A note on perinormal domains." J. Commut. Algebra 10 (3) 305 - 315, 2018. https://doi.org/10.1216/JCA-2018-10-3-305