A graded module over an ungraded ring 
is a module 
with a direct sum decomposition
into submodules. The index
set for the collection of submodules is called
the grading set.
For a graded ring 
indexed by the integers,
a graded module 
is a direct sum of Abelian
groups with 
.
Each 
is an 
-submodule, but need not be an 
-submodule (Stacks Project Authors
2026).
Graded modules arise naturally in homology. In particular, for every integer 
, there exists an 
th homology group 
of a topological space 
, and usually the "total homology"
of the topological space is considered to be
the direct sum of all the 
s. This makes the "total" homology of 
a module graded over the integers.
