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:

$A=\underset{i=0}{\overset{\mathrm{\infty }}{⨁}}{A}_i$

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