Exercise 27, § 1

In Exercise 22 take $\Gamma$ to be a direct sum of a family $(\Gamma_i)_{i \in I}$, where the $\Gamma_i$ are subgroups of the additive group $\mathbf{R}$. Show that the ring B defined in Exercise 22 (b) is a domain of finite character and real type and that it is the intersection of a family of essential valuation rings for B; moreover, every prime ideal $\neq 0$ of B is maximal.