Stability of some integral domains on a pullback

Abstract

Let $I$ be a nonzero ideal of an integral domain$T$ and let $\varphi\:T\to T/I$ be the canonicalsurjection. If $D$ is an integral domain contained in $T/I$, then$R=\varphi^{-1}\left(D\right)$ arises as a pullback of type$\square$ in the sense of Houston and Taylor such that $R\subseteqT$ is a domains extension. The stability of atomic domains,domains satisfying ACCP, HFDs, valuation domains, PVDs, AVDs,APVDs and PAVDs observed on all corners of pullback of type$\square$ under the assumption that the domain extension$R\subseteq T$ satisfies $Condition$ $1:$ For each $bın T$ thereexist $uın\cup(T)$ and $aın R$ such that $b=ua$.