A note on perinormal domains

Recently, N. Epstein and J. Shapiro introduced and studied the perinormal domains: those domains A whose going down overrings are flat A-modules. We show that every Pr\"ufer v-multiplication domain is perinormal and has no proper lying over overrings. We also show that a treed perinormal domain is a Pr\"ufer domain. We give two pull-back constructions that produce perinormal/non-perinormal domains.