Stability of some integral domains on a pullback

Abstract

Let $I$ be a nonzero ideal of an integral domain$T$ and let $\phi T\to T/I$ be the canonicalsurjection. If $D$ is an integral domain contained in $T/I$, then$R={\phi }^{-1}\left(D\right)$ arises as a pullback of type$◻$ in the sense of Houston and Taylor such that $R\text{subseteqT}$ 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$◻$ under the assumption that the domain extension$R\subseteq T$ satisfies $Condition$ $1:$ For each $bınT$ thereexist $uın\cup (T)$ and $aınR$ such that $b=ua$.