Example

(1) Let $B$ be a graded ring of type $\Delta$; if $B$ is given its canonical $\mathbf{Z}$-algebra structure (§ 1, no. 1, Example 2), $B$ is a graded $A$-algebra ($\mathbf{Z}$ being given the trivial graduation).

(2) Let $A$ be a graded commutative ring of type $\Delta$ and $M$ a graded $A$-module of type $\Delta$. Suppose that all the elements of the monoid $\Delta$ are cancellable, which allows (II, § 11, no. 6) us to define on $\mathrm{Homgr}_A(M, M) = \mathrm{Endgr}_A(M)$ a graded $A$-module structure of type $\Delta$; as this graduation is compatible with the ring structure on $\mathrm{Endgr}_A(M)$ (II, § 11, no. 6), it defines a unital graded $A$-algebra structure on the $A$-algebra $\mathrm{Endgr}_A(M)$.

(3) Algebra of a magma. Let $S$ be a magma and $\phi : S \to \Delta$ a homomorphism. For all $\lambda \in \Delta$, we write $S_\lambda = \phi^{-1}(\lambda)$; then $S_\lambda S_\mu \subset S_{\lambda + \mu}$. Let $A$ be a graded commutative ring of type $\Delta$ and $(A_\lambda)_{\lambda \in \Delta}$ its graduation; we shall define a graded $A$-algebra structure on the algebra $E = A^{(S)}$ of the magma $S$ (§ 2, no. 6). To this end, let $E_{\lambda}$ denote the additive subgroup of $E$ generated by the elements of the form $\alpha.s$ such that $\alpha \in A_{\mu}, s \in S_{\nu}$ and $\mu + \nu = \lambda$. As the $S_{\lambda}$ are pairwise disjoint, $E$ is the direct sum of the $A_{\mu}S_{\nu}$ and hence also the direct sum of the $E_{\lambda}$ and it is immediate that the $E_{\lambda}$ satisfy conditions (1) and (2) and therefore define on $E$ the desired graded $A$-algebra structure. If $S$ admits an identity element $e$, it may also be supposed that $\phi(e) = 0$. A particular case is the one where the graduation of the ring $A$ is trivial; then $E_{\lambda}$ is the sub-$A$-module of $E$ generated by $S_{\lambda}$. More particularly, if we take $S = \mathbf{N}^{(I)}, \Delta = \mathbf{N}$ and $\phi$ the mapping such that $\phi((n_i)) = \sum_{i \in I} n_i$, the ring $A$ having the trivial graduation, a graduation is thus obtained on the polynomial algebra $A[X_i]_{i \in I}$, for which the degree of a homogeneous polynomial $\neq 0$ is the total degree defined in § 2, no. 9 (cf. § 6, no. 6).

We now take $S$ to be the free monoid $Mo(B)$ of a set $B$ (I, § 7, no. 2) and $\phi$ the homomorphism $Mo(B) \to \mathbf{N}$ which associates with each word its length. Thus a graded $A$-algebra structure is obtained on the free associative algebra of the set $B$ (§ 2, no. 7; cf. § 5, no. 5).