Corollary 2

If N is a finitely generated submodule of M, N admits a finite generating system consisting of homogeneous elements.

It suffices to apply Corollary 1 noting that an element of M has only a finite number of homogeneous components $\neq 0$.

A graded submodule of $A_s$ (resp. $A_d$) is called a graded left (resp. right) ideal of the graded ring A. For every subring B of $A(B \cap A_\lambda)(B \cap A_\mu) \subset B \cap A_{\lambda+\mu}$; if B is a graded sub-$\mathbf{Z}$-module of A, the graduation induced on B by that on A is therefore compatible with the ring structure on B; B is then called a graded subring of A.

Clearly if N (resp. B) is a graded sub-A-module of M (resp. a graded subring of A), the canonical injection $N \to M$ (resp. $B \to A$) is a graded module homomorphism of degree 0 (resp. a graded ring homomorphism).

If N is a graded submodule of a graded A-module M and $(M_\lambda)_{\lambda \in \Delta}$ the graduation of M, the submodules $(M_\lambda + N)/N$ of $M/N$ form a graduation compatible with the structure of this quotient module. For, if $N_\lambda = M_\lambda \cap N$, $(M_\lambda + N)/N$ is identified with $M_\lambda/N_\lambda$ and it follows from Proposition 2 and § 1, no. 6, formula (26) that $M/N$ is their direct sum. Moreover,

and hence $A_\lambda((M_\mu + N)/N) \subset (M_{\lambda+\mu} + N)/N$, which establishes our assertion.

tion. The graduation ((M_\lambda + N)/N)_{\lambda \in \Delta} is called the quotient graduation of that on M by N and the quotient module M/N with this graduation is called the graded quotient module of M by the graded submodule N; the canonical homomorphism M \to M/N is a graded homomorphism of degree 0 for this graduation.

If b is a graded two-sided ideal of A, the quotient graduation on A/b is compatible with the ring structure on A/b; the ring A/b with this graduation is called the quotient graded ring of A by b; the canonical homomorphism A \to A/b is a homomorphism of graded rings for this graduation.