Exercise 1, § 3

Under the hypotheses of Proposition 1 of no. 1, show that every prime ideal $p \in \operatorname{Ass}_f(M)$ (§ 2, Exercise 17) is graded and is a minimal element of the set of prime ideals containing the annihilator of a homogeneous element of $M$. (Show first that the largest graded ideal $p'$ contained in a prime ideal $p$ of $A$ is prime, observing that, if the product of two elements $x, y$ belongs to $p'$, so does the product of their homogeneous components $\neq 0$ of highest degree. Note then that, if a prime ideal contains the annihilator of an element $z \in M$, it also contains the annihilator of at least one of the homogeneous components $\neq 0$ of $z$.)