Exercise 11, § 1

Let $A$ be a ring, $M$ an $A$-module, $x_1,\ldots,x_r$ elements of $A$. (a) For every integer $p$ such that $1 \leq p \leq r$, put $M_p = M/(x_2M+\ldots+x_pM)$. Prove that if the sequence $(x_1,\ldots,x_r)$ is $M$-regular, the sequence $(x_1,x_{p+1},x_{p+2},\ldots,x_r)$ is $M_p$-regular (reason by induction on $p$). (b) In order that the sequences $(x_1,x_3)$ and $(x_2,x_3)$ be $M$-regular, it is necessary and sufficient that the sequence $(x_1x_2,x_3)$ be $M$-regular. (c) Let $x'_p \in A$ $(1 \leq p \leq r)$. In order that the sequences $(x_1,\ldots,x_p,\ldots,x_r)$ and $(x_1,\ldots,x'_p,\ldots,x_r)$ be $M$-regular, it is necessary and sufficient that the sequence $(x_1,\ldots,x_p x'_p,\ldots,x_r)$ be $M$-regular (use $a$ and $b$). (d) Let $n_1,\ldots,n_r$ be integers $\geq 1$. In order that the sequence $(x_1^{n_1},\ldots,x_r^{n_r})$ be $M$-regular, it is necessary and sufficient that the sequence $(x_1,\ldots,x_r)$ be $M$-regular.