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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?

user26857
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selfstudying
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1 Answers1

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If there were any homomorphism $\phi:D[[S]]\to A$, then $\phi(1/n)$ would be an inverse for $n$ in $A$, but we are assuming $n$ is not invertible in $A$. For an explicit example, consider $A=\mathbb{Z}$ with $I=0$.

Eric Wofsey
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