Exercise 19, § 8

a) Let $n$ be an integer $\geqslant 0$, $(M_\alpha)_{\alpha \in I}$ an inductive system of $A$-modules, $M$ its limit. Assume that $\mathrm{Card}(I) \leqslant \aleph_n$ (E, III, p. 87, Exercise 10) and that there exists an integer $r$ such that $\mathrm{dp}_A(M_\alpha) \leqslant r$ for every $\alpha \in I$. Now prove that one has $\mathrm{dp}_A(M) \leqslant r + n + 1$ (reason by induction on $n$, using Exercise 18, a) and Exercise 17).

b) Assume that every left ideal of $A$ is generated by a family of elements of cardinal $\leqslant \aleph_n$. Prove that one has $\mathrm{dp}_A(M) \leqslant \mathrm{dpl}_A(M) + n + 1$ for every $A$-module $M$ generated by a family of elements of cardinal $\leqslant \aleph_n$. Deduce the inequalities