If R is a local ring then R[[x]] is local.

Recall that if S is a commutative ring with unit, S is local if and only the set of non unit of S is an ideal, i.e, denoting with S* the unit of S, S-S* is an ideal of S.

Now, let m be the maximal ideal of R. We prove that R[[x]] – R[[x]]* = (m,x). Indeed if f is a formal power series with coefficient in R and is not invertible, we have that f(0) is not invertible (because in general ∑ aixⁱ ∈ R[[x]] is invertible iff a0 is invertible). Then f(0) is contained in the only maximal ideal of R, i.e f(0)∈m, so f=f(0)+xg ∈ (m,x). Viceversa if f∈(m,x) then f(0)∈m so f(0) is not invertible and so f∈R[[x]]_R[[x]]*. □