##### Actions

An algebra $A$ whose additive group can be represented in the form of a (weak) direct sum of groups $A _ {i}$, $i = 0, 1 \dots$ where $A _ {i} A _ {j} \subseteq A _ {i+} j$ for any $i, j$. As a result, the additive group of a graded algebra (considered as a module over the ring of integers) is a positively graded module. As an example of a graded algebra one can quote the algebra $A = F [ x ]$ of polynomials over a field $F$, where $A _ {i}$ is the subspace generated by the monomials of degree $i$( $A _ {0} = F$). One can also define a graded algebra $A$ in a more general manner as an algebra whose additive group can be represented as a direct sum of groups $A _ \alpha$, where $\alpha$ runs through a certain commutative semi-group $G$ and $A _ \alpha A _ \beta \subseteq A _ {\alpha + \beta }$ for any $\alpha , \beta \in G$. The concept of a filtered algebra is closely connected with that of a graded algebra. It is in fact possible to define, in a natural manner, an ascending filtration

$$A = \cup _ {k \geq 0 } \mathfrak A _ {k} ,\ \ \mathfrak A _ {0} \subset \mathfrak A _ {1} \subset \dots ,\ \ \mathfrak A _ {k} = \ \sum _ {i = 0 } ^ { k } A _ {i} ,$$

on each graded algebra $A = \sum _ {i \geq 0 } A _ {i}$. Conversely, if $A = \cup _ {k \geq 0 } \mathfrak A _ {k}$ is a filtered algebra $( \mathfrak A _ {0} \subset \mathfrak A _ {1} \subset \dots, \mathfrak A _ {i} \mathfrak A _ {j} \subset \mathfrak A _ {i+} j )$, then one defines the graded algebra $\mathop{\rm gr} A = GA = \sum _ {i \geq 0 } A _ {i}$( where $A _ {i} = \mathfrak A _ {i} / \mathfrak A _ {i-} 1$, $A _ {0} = \mathfrak A _ {0}$), which is called the graded algebra associated with $A$. A graded ring is defined in a similar manner.

For an arbitrary group $G$ one may define a gradation of type $G$ on an algebra $A$, i.e. $A = \oplus _ {\sigma \in G } A _ \sigma$ where each $A _ \sigma$ is an additive subgroup of $A$ and $A _ \sigma A _ \tau \subset A _ {\sigma + \tau }$ for all $\sigma , \tau \in G$. The group algebra $k G$ over the field $k$, as well as crossed products $k \star G$ defined by a group morphism $\Phi : G \rightarrow \mathop{\rm Aut} ( k)$ and a $2$- cocycle $c \in H ^ {2} ( G, k ^ {*} )$, are examples of $G$- graded algebras (see also Cross product). The use of $\mathbf Z$- gradations that are not necessarily positive gradations allows one to consider the graded rings associated to $I$- adic filtrations on a ring $R$; for an ideal $I$ of $R$ the $I$- adic filtration is given by a descending chain $R \supset I \supset I ^ {2} \supset \dots \supset I ^ {n} \supset \dots$, and then $G ( R) = \oplus _ {n \in \mathbf N } I ^ {n} / I ^ {n+} 1$, where $G ( R) _ {-} n = I ^ {n} / I ^ {n+} 1$ is negatively graded.