Exercise 1, § 2

Let $(C_n, \pi_{n,m})$ be a projective system of rings relative to the set of indices $\mathbf{N}$. Suppose that $C_n$ is artinian for every $n \in \mathbf{N}$ and that the homomorphisms $\pi_{n,m}$ are surjective. Let $\pi_n$ be the canonical homomorphism of $C = \varprojlim C_n$ into $C_n$. Now prove that for every $x \in C$, one has $xC = \varprojlim \pi_n(x)\ C_n$. (Reason as in the proof of Prop. 3 of § 2, No. 1.)

EXERCISES

AC IX.69