Let $R$ be a commutative ring with identity. Recall that an idempotent element $e$ of $R$ is an element a such that $e^2=e$, and a local idempotent is an idempotent a such that $Re$ is a local ring. We know the following facts:
1) $f\in R[[x]]$ is idempotent if and only if $f^2=f\in R$, where $R[[x]]$ is the ring of power series over R.
2) The maximal ideals of the ring $R[[x]]$ have the form $M’=(M,x)$ where $M$ is a maximal ideal of $R$.
Question: Let $e^2=e$ be an idempotent element in $R$ (also in $R[[x]]$) if the following is true?: $e^2=e$ is a local idempotent element in $R$ if and only if $e^2=e$ is a local idempotent element in $R[[x]]$.
Yes, this is true. By fact #1, you've shown that the idempotents of $R$ and $R[[x]]$ are exactly the same, so all that's left to do is to look at the locality condition. As $(R[[x]])e = (Re)[[x]]$, and the latter has exactly one maximal ideal because of statement #2, your claim follows.