A decomposition of a module into a direct sum of submodules. The index set for the collection of submodules is then called the grading set.
Graded modules arise naturally in homology. In particular, for every integer 
,
there exists an 
th
homology group of a space 
, and usually the "total homology" of the space
is considered to be the direct sum of all the 
s. This makes the "total" homology of 
a module graded over the integers.
