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.