Exercise 5, § 9

Let $A$ be a ring, $M$ an a-module, $x = (x_1, ..., x_n)$ a sequence of elements of A, $k = (k_1, ..., k_n) \in \mathbf{N}^n$, $x^k = (x_1^{k_1}, ..., x_n^{k_n})$. Prove that the sequence $x^k$ is M-regular if and only if the sequence x is so; in this case the a-module $M/(x^k)$ M admits a composition sequence whose quotients are isomorphic to $M/(x)$ M. (Reason by induction on n.)