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.