Linkage of ideals over a module

Inspired by the works in linkage theory of ideals, we define the concept of linkage of ideals over a module. Several known theorems in linkage theory are improved or recovered by new approaches. Specially, we make some extensions and generalizations of the basic result of Peskine and Szpiro \cite[prop 1.3]{PS}, namely if $R$ is a Gorenstain local ring, $\mathfrak{a} \neq 0$ (an ideal of $R$) and $\mathfrak{b} := 0:_R \mathfrak{a}$ then $\frac{R}{\mathfrak{a}}$ is Cohen-Macaulay if and only if $\frac{R}{\mathfrak{a}}$ is unmixed and $\frac{R}{\mathfrak{b}}$ is Cohen-Macaulay.