##### Actions

A module $A$ that can be represented as the direct sum of its submodules $A _ {n}$( the index $n$ runs through all integers; some of the submodules $A _ {n}$ may be trivial). A module $A$ is called positively graded if $A _ {n} = 0$ for all $n < 0$, and negatively graded if $A _ {n} = 0$ for all $n > 0$. The non-zero elements of $A _ {n}$ are called homogeneous elements of degree $n$. A submodule $B$ of a graded module $A$ is said to be homogeneous if it can be decomposed into a direct sum of submodules $B _ {n}$ such that $B _ {n} \subseteq A _ {n}$ for any integer $n$; $B$ is then a graded module. If $B$ is a homogeneous submodule of a graded module $A$, then the quotient module $\overline{A}\; = A/B$ is also a graded module, i.e. $\overline{A}\; = \sum \overline{A}\; _ {n}$, where $\overline{A}\; _ {n}$ is the image of the submodule $A _ {n}$ under the natural homomorphism $A \rightarrow A/B$, $\overline{A}\; _ {n} \simeq A _ {n} /B _ {n}$. Graded modules are extensively used in homological algebra.