Definition 1

Let $\Delta$ be a commutative monoid, $A$ a graded ring of type $\Delta$ (II, § 11, no. 2), $(A_\lambda)_{\lambda \in \Delta}$ its graduation and E an A-algebra. A graduation $(E_\lambda)_{\lambda \in \Delta}$ of type $\Delta$ on the additive group E is said to be compatible with the A-algebra structure on E if it is compatible both with the A-module and with the ring structure on E, in other words, if, for all $\lambda, \mu$ in $\Delta$,

(1)

(2)

The A-algebra E, with this graduation, is then called a graded algebra of type $\Delta$ over the graded ring A.

When the graduation on A is trivial (that is (II, § 11, no. 1) $A_0 = A, A_\lambda = \{0\}$ for $\lambda \neq 0$), condition (1) means that the $E_\lambda$ are sub-A-modules of E. This leads to the definition of the notion of graded algebra of type $\Delta$ over a non-graded commutative ring $A$: $A$ is given the trivial graduation of type $\Delta$ and the above definition is applied.

When we consider graded $A$-algebras $E$ with a unit element $e$, it will always be understood that $e$ is of degree 0 (cf. Exercise 1).

It follows that if an invertible element $x \in E$ is homogeneous and of degree $p$, its inverse $x^{-1}$ is homogeneous and of degree $-p$: it suffices to decompose $x^{-1}$ as a sum of homogeneous elements in the relations $x^{-1}x = xx^{-1} = e$.

Let $E$ and $E'$ be two graded algebras of type $\Delta$ over a graded ring $A$ of type $\Delta$. An $A$-algebra homomorphism $u : E \to E'$ is called a graded algebra homomorphism if $u(E_\lambda) \subset E'_\lambda$ for all $\lambda \in \Delta$ (where $(E_\lambda)$ and $(E'_\lambda)$ denote the respective graduations of $E$ and $E'$); where $E$ and $E'$ are associative and unital and $u$ is unital, this condition means that $u$ is a graded ring homomorphism (II, § 11, no. 2).

Let $E$ be a graded $A$-algebra of type $N$. $E$ is identified with a graded $A$-algebra of type $\mathbf{Z}$ by writing $E_n = \{0\}$ for $n < 0$.