Example

(1) On any ring $A$ the trivial graduation of type $\Delta$ is compatible with the ring structure. If $A$ is graded by the trivial graduation, for a graduation $(M_\lambda)$ of type $\Delta$ on an $A$-module $M$ to be compatible with the $A$-module structure, it is necessary and sufficient that the $M_\lambda$ be submodules of $M$.

(2) Let $A$ be a graded ring of type $\Delta$, $M$ a graded $A$-module of type $\Delta$ and $\rho$ a homomorphism of $\Delta$ into a commutative monoid $\Delta'$ whose identity element is denoted by 0. Then $A$ is a graded ring of type $\Delta'$ and $M$ a graded module of type $\Delta'$ for the graduations of type $\Delta'$ derived from $\rho$ and the graduations of type $\Delta$ on $A$ and $M$ by the procedure of no. 1, Example 1: this follows immediately from the relation $\rho(\lambda + \mu) = \rho(\lambda) + \rho(\mu)$.

In particular, if $\Delta = \Delta_1 \times \Delta_2$ is a product of two commutative monoids, the projections $\mathrm{pr}_1$ and $\mathrm{pr}_2$ are homomorphisms and the corresponding graduations are just the partial graduations derived from the graduations of type $\Delta$ (no. 1, Example 3); these partial graduations are thus compatible with the ring structure of $A$ and the module structure of $M$.

Similarly, if $\Delta = \Delta_0^{(r)}$ (where $\Delta_0$ is a commutative monoid with identity element denoted by 0), the total graduation (no. 1, Example 4) of type $\Delta_0$ derived from the graduation of type $\Delta$ on $A$ (resp. $M$) by means of the codiagonal homomorphism is compatible with the ring structure on $A$ (resp. with the module structure on $M$).

(3) Let $A$ be a graded ring of type $\Delta$, $M$ a graded $A$-module of type $\Delta$ and $\lambda_0$ an element of $\Delta$; for all $\lambda \in \Delta$, let $M'_\lambda = M_{\lambda + \lambda_0}$ and let $M'$ be the $\mathbf{Z}$-module $\bigoplus_{\lambda \in \Delta} M'_\lambda$. As $A_\lambda M'_\mu \subset M_{\lambda + \mu + \lambda_0} = M'_{\lambda + \mu}$, $M'$ is an $A$-module and the $M'_\lambda$ form on $M'$ a graduation of type $\Delta$ compatible with the $A$-module structure of $M'$; the graded $A$-module $M'$ of type $\Delta$ thus defined is said to be obtained by shifting by $\lambda_0$ the graduation of $M$ and it is denoted by $M(\lambda_0)$. When $\Delta$ is a group, the underlying $A$-module of the graded $A$-module $M'$ is identified with $M$.

(4) Let $B$ be a commutative ring. The polynomial ring $B[X]$ in one indeterminate is graded of type $\mathbf{N}$ by the subgroups $BX^n$ ($n \geqslant 0$) (cf. III, § 2, no. 9 and IV).

(5) Let $B$ be a commutative ring, $E$ a $B$-module, $Q$ a quadratic form on $E$ and $C(Q)$ the Clifford algebra of $Q$ (cf. IX, § 9). The sub-$B$-modules $C^+(Q)$ and $C^-(Q)$ form on $C(Q)$ a graduation of type $\mathbf{Z}/2\mathbf{Z}$ compatible with the ring structure on $C(Q)$.