Abstract: As a proper generalization of D4 modules, the concept of generalized D4 modules is introduced in this paper. Some equivalent conditions of generalized D4 modules are given by studing the basic proerties of such modules. Furthermore, the concept of pseudo D4 modules is introduced. and it is shown that a ring R is semisimple if and only if each R-module is a pseudo D4 module.

Key words: generalized D4 module, semisimple ring, direct summand, pseudo D4 module