Exercise 2, § 3

In a graded ring $A$ of type $A$ (where $A$ is torsion-free) let $p$ be a minimal element of the set of prime ideals of $A$, which is necessarily graded (Exercise 1). Show that for every homogeneous element $t \in p$ there exists a homogeneous element $s \notin p$ and an integer $n > 0$ such that $st^n = 0$. Deduce generalizations of Propositions 3, 4 and 5 to graded rings which are not necessarily Noetherian (with the definitions of § 2, Exercises 12 and 20).