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. Specially, we make some extensions and generalizations of a basic result of Peskine and Szpiro cite[Proposition 1.3]{PS}, namely if $R$ is a Gorenstein local ring, $ a neq 0$ (an ideal of $R$) and $ b := 0:_R a$ then $frac{R}{a}$ is Cohen-Macaulay if and only if $frac{R}{a}$ is unmixed and $frac{R}{ b}$ is Cohen-Macaulay.