On Graded Second and Coprimary Modules and Graded Secondary Representations

In this paper we introduce and study the concepts of graded second (gr-second) and graded coprimary (gr-coprimary) modules which are different from second and coprimary modules over arbitrary-graded rings. We list some properties and characterizations of gr-second and gr-coprimary modules and also study graded prime submodules of modules with gr-coprimary decompositions. We also deal with graded secondary representations for graded injective modules over commutative-graded rings. By using the concept of sigma-suspension (sigma)M of a graded module M, we prove that a graded injective module over a commutative graded Noetherian ring has a graded secondary representation.