Exercise 17, § 8

Let $M$ be an $A$-module, $I$ an ordered set, $(M_\alpha)_{\alpha \in I}$ an increasing family of submodules of $M$, such that $M = \bigcup_{\alpha \in I} M_\alpha$. We put $M'_\alpha = \bigcup_{\beta < \alpha} M_\beta$ for $\alpha \in I$.

a) Exercise prove the inequality $\mathrm{dp}_A(M) \leqslant \sup_{\alpha \in I} \mathrm{dp}_A(M_\alpha/M'_\alpha)$.

b) Suppose that there exists an integer $n$ such that $\mathrm{dp}_A(M'_\alpha) \leqslant n$ for all $\alpha \in I$. Now prove that one has