Remark

When $A$ has the trivial graduation (no. 1, Example 1), $(M \otimes_A N)_\lambda$ is then simply the direct sum of the sub-$Y$-modules $M_\mu \otimes_A N_\nu$ of $M \otimes_A N$ such that $\mu + \nu = \lambda$.

Let $M$ (resp. $N$) be a graded right (resp. left) $A$-module of type $\Delta$, $P$ a graded $\mathbf{Z}$-module of type $\Delta$ and let $f$ be a $\mathbf{Z}$-bilinear mapping of $M \times N$ into $P$ satisfying condition (1) of $§ 3$, no. 1, and such moreover that

Then $f(x, y) = g(x \otimes y)$, where $g : M \otimes_A N \to P$ is a $\mathbf{Z}$-linear mapping ($§ 3$, no. 1, Proposition 1) and it follows from the above condition that $g$ is a graded $\mathbf{Z}$-module homomorphism of degree 0.

Let $B$ be another graded ring of type $\Delta$ and $\rho : A \to B$ a homomorphism of graded rings (no. 2); then $\rho^*(B_d)$ is a graded right $A$-module of type $\Delta$. If $E$ is a graded left $A$-module of type $\Delta$ and $\rho^*(B_d) \otimes_A E$ is given the graded $\mathbf{Z}$-module structure of type $\Delta$ defined above, the canonical left $B$-module structure ($§ 5$, no. 1) is compatible with the graduation of

The graded $B$-module thus obtained is said to be obtained by extending the ring of scalars to $B$ by means of $\rho$ and when we speak of $E_{(B)}$ or $\rho^*(E)$ as a graded $B$-module, we always mean this structure, unless otherwise mentioned.