Graded Modules Over First Strongly Graded Rings

Let g be a group with identity e and r =⊕g ∈ g rg be a g-graded ring. we say r is first strongly graded if rgrh = rgh,for every g; h ∈ supp(r;g). in this paper,we investigate several properties of group-graded modules over first strongly graded rings concerning annihilators and multiplication modules,semisimple modules,and other aspects of modules. for example,we prove that the nonzero components of a graded module over a commutative first strongly graded ring whose degrees lie in supp(r;g) possess an equal annihilators over r. adding an extra condition,we can further partition the group g into three parts such that the annihilators of the components of the graded module whose degrees lying in the same part possess the same annihilator. also,we show that over first strongly graded rings the gr-semisimple r-modules are only the flexible r-modules with semisimple e-components.