Exercise 13, § 1

Let K be a field, $(n_r)_{r \geq 1}$ a strictly increasing sequence of integers $> 0$; for every integer $i \geq 0$, let $p_i$ denote the ideal of the ring $K[(X_j)_{j \in \mathbf{N}}]$ generated by the $X_j$ for $n_1 + \cdots + n_i \leq j < n_1 + \cdots + n_i + n_{i+1}$, and let S be the set of elements of $K[(X_j)]$ which belong to none of the $p_i$. Then the ring $S^{-1}K[(X_j)]$ is noetherian and of infinite dimension (use Exercise 12 to show that $S^{-1}K[(X_j)]$ is noetherian) $^1$.