Are there any hints to solve the exercise 1.3.9(c) in Qing Liu's book Algebraic Geometry and Arithmetic Curves?
Let $n\geq 2$ be an integer, and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Let $x\in I$. How can I show that if $n$ is not invertible in $A$, then there does not necessary exists a unique continuous homomorphism $\phi:D[[S]]\to A$ such that $\phi(S)=x$.
I think the idea is that if we take the unique ring homomorphism $\mathbb Z\to A$ then it does not necessary extend to a unique ring homomorphism $D\to A$. But what would be an explicit counterexample?