$Gr$-injective modules and $gr$-projective modules over $G$-graded commutative rings

It is well known that the decomposition of injective modules over noetherian rings and the decomposition of projective modules over artinian rings are among the most beautiful and important results in commutative algebra. Our aim is to prove similar results for graded rings. It is important for us to understand the structure of the modules over the graded rings. In this paper, we study the structure theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings and the structure theorem for $gr$-projective modules over $gr$-artinian $G$-graded commutative rings.