Corollary

For every (necessarily graded) prime ideal $p$ associated with a graded $A$-module $M$, there exists an index $k \in A$ such that the graded $A$-module $(A/p)(k)$ obtained from the graded $A$-module $A/p$ by diminishing the degrees by $k$ (Algebra, Chapter II, § 11, no. 2) is isomorphic to a graded submodule of $M$.

With the notation of the proof of Proposition 1, consider the homomorphism obtained, by taking quotients, from the homomorphism $a \mapsto ax_{i(n)}$ of $A$ to $M$; the latter is a graded homomorphism of degree $i(n)$ and hence it gives on taking quotients a graded bijective homomorphism of degree $i(n)$ of $A/p$ onto a graded submodule of $M$.