Exercise 2, § 2

Let A be a ring and $(J_n)_{n \in \mathbf{N}}$ a decreasing sequence of ideals of A, such that $pJ_n + J_n^p \subset J_{n+1}$ for all $n \in \mathbf{N}$. a) Prove that the statements of lemma 1 and of prop. 1 of § 1, No. 1 are still true under this assumption. b) Let i and n be two positive integers. When $i \leq n$ the mapping $x \mapsto p^i x^{p^{n-i}}$ of A into A defines, by passing to the quotients, a mapping

For $i > n$, one sets $\rho_{n,i}^A = 0$. For $x \in A/J_n$ and $a \in A/J_0$, one has $x^{p^{n-i}} \rho_{n,i}^A(a) = \rho_{n,i}^A(\overline{x}a)$, where $\overline{x}$ is the image of x in $A/J_0$. If j is a positive integer and if $a, b$ are two elements of $A/J_n$, one has c) Let $(R_n)_{n \in \mathbf{N}}$ be the sequence of polynomials constructed in exerc. 3, p. 42. Let n and i be two positive integers, $a$ and $b$ two elements of $A/J_0$. Then one has, in $A/J_n$, the equality