 Formal power series over a local ring

If $(A,\mathfrak{m})$ is a local ring, then $B=A[[x]]$ is local with maximal ideal $I=\mathfrak{m}B+xB$ and residue field $A/\mathfrak{m}$. A series $f=\sum a_n{x}^n$ is a unit if and only if $a_0$ is a unit. Necessity follows by taking constant coefficients of an inverse; sufficiency follows from the recursion $b_0=a_0^{-1}$ and $b_n=-a_0^{-1}{\sum }_{i=1}^n{a}_i{b}_{n-i}$ for its inverse. Thus the nonunits are precisely series with $a_0\in \mathfrak{m}$, which is exactly $I$, since $f=a_0+xg$. The constant-coefficient map followed by reduction modulo $\mathfrak{m}$ has kernel $I$ and image $A/\mathfrak{m}$.