The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains, \(\phi \) -Prüfer rings, Bézout domains, and \(\phi \) -Bézout rings. Let \(G\mathcal {H}=\{R \mid R\) is a commutative ring admitting a divided prime ideal \(P \subseteq Z(R) \}\) . Let \(R \in G\mathcal {H}\) and T(R) be the total ring of quotients of R. Define \(\phi _P: T(R) \longrightarrow R_{P}\) by \(\phi (a / b)=a / b\) for every \(a \in R\) and \(b \in R \setminus Z(R)\) . Then \(\phi _P\) is a ring homomorphism from T(R) into \(R_{P}\) , and \(\phi _P\) restricted to R is also a ring homomorphism from R into \(R_{P}\) given by \(\phi _P(x)=x / 1\) for every \(x \in R\) . If \(Z(\phi _P(R))=\phi _P(P)\) , then R is called a strongly \(\phi _P\) -ring. A P-ideal I of R (i.e., \(P \subset I\) ) is said to be \(\phi _P\) -invertible if \(\phi _P(I)\) is an invertible ideal of \(\phi _P(R)\) . If every finitely generated P-ideal of R is \(\phi _P\) -invertible, then we say that R is a \(\phi _P\) -Prüfer ring. We also say that R is a \(\phi _P\) -Bézout ring if \(\phi _P(I)\) is a principal ideal of \(\phi _P(R)\) for every finitely generated P-ideal I of R. We show that the theories of \(\phi _P\) -Prüfer and \(\phi _P\) -Bézout rings are similar to those of Prüfer and Bézout domains.
