## Abstract

In this paper, we introduce some classes of *R*-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let *R* be a commutative ring with identity and set

$\mathbb{H}=\{M\mid M\text{is an}R\text{-module and}\mathrm{Nil}(M)\text{is a divided prime submodule of}M\}.$

For an *R*-module $M\in \mathbb{H}$, set $T=(R\setminus Z(R))\cap (R\setminus Z(M))$, $\U0001d517(M)={T}^{-1}M$ and $P=(\mathrm{Nil}(M){:}_{R}M)$. In this case, the mapping $\mathrm{\Phi}:\U0001d517(M)\to {M}_{P}$ given by $\mathrm{\Phi}(x/s)=x/s$ is an *R*-module homomorphism. The restriction of Φ to *M* is also an *R*-module homomorphism from *M* into ${M}_{P}$ given by $\mathrm{\Phi}(x)=x/1$ for every $x\in M$. A nonnil submodule *N* of *M* is said to be Φ-invertible if $\mathrm{\Phi}(N)$ is an invertible submodule of $\mathrm{\Phi}(M)$. Moreover, *M* is called a Φ-Prüfer module if every finitely generated nonnil submodule of *M* is Φ-invertible. If every nonnil submodule of *M* is Φ-invertible, then we say that *M* is a Φ-Dedekind module. Furthermore, *M* is said to be a Φ-Bezout module if $\mathrm{\Phi}(N)$ is a principal ideal of $\mathrm{\Phi}(M)$ for every finitely generated submodule *N* of the *R*-module *M*. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout *R*-modules.

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