Domains whose ideals meet a universal restriction

Let $S(D)$ represent a set of proper nonzero ideals $I(D)$ (resp., $t$ -ideals $I_t(D)$) of an integral domain $D\mathrm{\ne }qf(D)$ and let $P$ be a valid property of ideals of $D\mathrm{.}$ We say $S(D)$ meets $P$ (denoted $S(D)⊲P)$ if each $s\in S(D)$ is contained in an ideal satisfying $P$. If $S(D)$ $⊲P,$ $\dim (D)$ can’t be controlled. When $R=D[X],$ $I(D)$ $⊲P$ does not imply $I(R)$ $⊲P$ while $I_t(D)$ $⊲P$ implies $I_t(R)$ $⊲P$ usually. We say $S(D)$ meets $P$ with a twist $($ written $S(D){⊲}^{t}P)$ if each $s\in S(D)$ is such that, for some $n\in N,$ $s^n$ is contained in an ideal satisfying $P$ and study $S(D){⊲}^{t}P,$ as its predecessor. A modification of the above approach is used to give generalizations of Almost Bezout domains.