Exercise 8, § 6

Let $A$ be a noetherian ring, and let $B$ be a graded $A$-subalgebra of the polynomial ring $A[T]$.

a) Prove that one has the inequalities

b) Let $B_i$ be the set of $b \in A$ such that $bT^i \in B$. Prove that if one has $\dim(B) = \dim(A)$, there exists a positive integer $k$ such that for every $i \geq k$, $B_i$ is contained in the intersection of the minimal prime ideals of A.