Document Type : Original Manuscript


1 Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8- 8349, Iran.

2 Department of Mathematics, Payame Noor University, Tehran, 19395-3697, Iran.


‎We introduce a generalization of the notion of‎ depth of an ideal on a module by applying the concept of‎ local cohomology modules with respect to a pair‎ ‎of ideals‎.
‎We also introduce the concept of $(I,J)$-Cohen--Macaulay modules as a generalization of concept of Cohen--Macaulay modules‎. ‎These kind of modules are different from Cohen--Macaulay modules‎, as an example shows‎. ‎Also an artinian result for such modules is given‎.