Graded Abelian group

Definition
A graded Abelian group is an Abelian group $$A$$ with a decomposition as a direct sum of subgroups $$A_i$$ where $$i$$ ranges over the nonnegative integers. In symbols:

$$A = \bigoplus_{i=0}^\infty A_i$$

Such an additional structure on an Abelian group is termed a gradation.