Graded module

Definition
Let $$A$$ be a graded ring with the $$i^{th}$$ graded component denoted by $$A_i$$. An $$A$$-graded module is a graded Abelian group $$M$$ that is also an $$A$$-module, satisfying the following additional condition:

$$A_iM_j \subset M_{i+j}$$