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

We introduce a notion of generalized local cohomology modules with respect to a pair of ideals $(I,J)$ which is a generalization of the concept of local cohomology modules with respect to $(I,J).$ We show that generalized local cohomology modules $H^i_{I,J}(M,N)$ can be computed by the v{C}ech cohomology modules. We also study the artinianness of generalized local cohomology modules $H^i_{I,J}(M,N).$