Let $R$ be a ring, and $S$ be a subring of $R$.
Denote $S[[x]]$ for a ring of formal power series with coefficients in $S$.
Let $\alpha \in R$ be a unit, such that $\alpha \notin S$.
Can there exist a ring homomorphism $\phi :S[[x]]->R$ such that $\phi $ sends $x\in S[[x]]$ to $\alpha \in R$?
If this question is hard to answer in general, could you answer for specific case?
For example, let R be Complex Numbers, S be integers, and $\alpha =6/5$.
EDIT: My question is restricting topology to the 'usual topology.'
Best Answer
About your example, we have an homomorphism $$\Bbb{Z}[[X]]\to \Bbb{Z}_2, \qquad \sum_n c_n X^n\to \sum_n c_n (6/5)^n$$ And $\Bbb{Q}_2$ is a field of characteristic $0$ with cardinality less than $\Bbb{C}$, by the axiom of choice it is isomorphic to a subfield of $\Bbb{C}$, thus obtaining your homomorphism $$\Bbb{Z}[[X]]\to \Bbb{Q}_2\to \Bbb{C},\qquad X\to 6/5$$