Generalizations of Prüfer rings and Bézout rings

Abstract

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.