Exercise 8, § 5

Let A be a ring and $n$ an integer $\geq 1$. Prove that the following conditions are equivalent: (i) One has $\mathrm{dh}(A) \leq n$. (ii) For every a-module M monogenic, one has $\mathrm{dp}_A(M) \leq n$. (iii) For every ideal I of A, one has $\mathrm{dp}_A(I) \leq n - 1$. (Use A, X, p. 138, Proposition 4 to prove the equivalence (i) $\Leftrightarrow$ (ii) and A, X, p. 93, Proposition 11 to prove (i) $\Leftrightarrow$ (iii).)