Stability of some integral domains on a pullback

Tariq Shah, Sadia Medhat

Let $I$ be a nonzero ideal of an integral domain $T$ and let $\varphi\:T\to T/I$ be the canonical surjection. If $D$ is an integral domain contained in $T/I$, then $R=\varphi^{-1}eft(D\right)$ arises as a pullback of type $\square$ in the sense of Houston and Taylor such that $R\subseteq T$ 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 $b\in T$ there exist $u\in\cup(T)$ and $a\in R$ such that $b=ua$.