Exercise 14, § 8

Let x be an element of A cancellable on the left. a) Show that the ideal $xA$ is two-sided if and only if there exists an endomorphism $\sigma$ of the ring A such that $ax = x\sigma(a)$ for all $a \in A$. b) We suppose from now on that the conditions of a) are verified. Let M be an A-module such that $xM = 0$. Prove the equality $\mathrm{dp}_A(M) = \mathrm{dp}_{A/xA}(M) + 1$ (one may use the spectral sequence of exercise 6, d), p. 188). c) We suppose that x belongs to the radical of A, and that A is Noetherian. Let N be a finitely generated A-module, such that the homothety of ratio x is injective in N. Show that one has $\mathrm{dp}_A(N) = \mathrm{dp}_{A/Ax}(N/xN)$. d) Under the conditions of c), show that one has $\mathrm{dh}(A) = \mathrm{dh}(A/Ax) + 1$.