Exercise 5, § 8

Let $(M_i)_{i \in E}$ be a family of A-modules. (a) Now prove that $\mathrm{di}_A(\prod_{i \in E} M_i) = \sup \mathrm{di}_A(M_i)$. (b) If A is noetherian, now prove that $\mathrm{di}_A(\bigoplus_{i \in E} M_i) = \sup \mathrm{di}_A(M_i)$. (c) If $\mathrm{di}_A(\bigoplus_{i \in E} M_i) = \sup \mathrm{di}_A(M_i)$ for every family $(M_i)_{i \in E}$ of A-modules, A is noetherian (use Exercise 21, p. 170).