I apologize if the question is too elementary. I did not get any response on math stackexchange.

I read the following statement in some algebraic topology notes and I want to know if it is true and, if so, why.

Let $R$ be a commutative ring and $f(x)$ a power series in $R[[x]]$. Suppose that $R[[x]]/(f)$ is a finitely generated $R$-module that is flat over $R$. Then there is some monic polynomial $\alpha(x)$ such that $f(x)$ and $\alpha(x)$ generate the same ideal in $R[[x]]$.

Suppose $f(x)=a_0+a_1x+a_2x^2+...$ I can show, without using flatness, that there is some n such that $(a_0,a_1,...,a_n)$ is the unit ideal in $R$. The claimed conclusion would follow if I knew that one of the $a_i$ is actually a unit. Is there some way to use flatness to show this? Alternatively, is there a counterexample?


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