I am trying to solve a problem in the book Introduction to commutative algebra from Atiyah.
Page 11, exercise 5 (iv). Let $A[\![x]\!]$ be the ring of all formal power series of the form $\sum_{i=0}^{\infty} a_ix^i$. It is said that the contraction $m^c$ of a maximal ideal $m$ of $A[\![x]\!]$ is a maximal ideal of $A$ and $m$ is generated by $m^c$ and $x$.
There is a counter example. Let $A=\mathbb{Z}$, $p$ be a prime and $m = \left\{ a_0+\sum_{i=1}^{\infty}a_ix^i \biggm| a_0 = kp, k\in \mathbb{Z}, a_i \in \mathbb{Z}\right\}.$ Then $m$ is a maximal ideal of $\mathbb{Z}[\![x]\!]$. The contraction $m^c$ of $m$ is $\{kp \mid k\in \mathbb{Z} \}$. The smallest ideal containing $m^c$ and $x$ is $K=\left\{\sum_{i=0}^{\infty}a_ix^i \biggm| a_i=k_i p, k_i \in \mathbb{Z}\right\}$ which is smaller than $m$. It seems that $m$ is not generated by $m^c$ and $x$. I don't know where is the problem. Thank you very much.