Exercise 25, § 1

Let $\Delta$ be a totally ordered commutative group. Extend Propositions 20 and 21 of no. 8 to graded rings of type $\Delta$ (for Proposition 20 use Exercise 24; for Proposition 21 generalize Lemma 4 of no. 8 to the case where $A$ is finitely generated and then prove Proposition 21 by taking the direct limit).