Exercise 1, § 2

Let $V$ be a discrete valuation ring, with field of fractions $K$, and let $n$ be an integer. Let $A$ be the subring of the polynomial ring $K[X_1, ..., X_n]$ consisting of the polynomials whose constant term belongs to $V$. The canonical homomorphism from $V$ to $A$ satisfies property (PM), and the canonical homomorphism $V/m_v \to A/m_vA$ is an isomorphism. Moreover, we have $\dim(A) \geq n + 1$ and $\dim(V) = 1$. As soon as $n \geq 1$, we therefore have $\dim(A) > \dim(V) + \dim(A/m_vA)$. Now prove that $A$ is not noetherian. Using formal series, construct an analogous example in which $A$ is local and non-noetherian.