Some properties of generalized local homology modules with respect to a pair of ideals

We introduce in this paper the definition of the i-th generalized local homology module with respect to a pair of ideals, and this module is a generalization of the i-th generalized local homology module with respect to an ideal, and thus we study some of its properties. Moreover, we presented a generalization of the notion of width of an ideal on a module, and also a generalization of the concept of a co-Cohen–Macaulay module.