= Formal power series over a local ring
{title2=$(A[[x]],\mathfrak mA[[x]]+(x))$}
If $(A,\mathfrak m)$ is a <local ring>, then $B=A[[x]]$ is local with <maximal ideal> $I=\mathfrak mB+xB$ and <residue field> $A/\mathfrak m$. A series $f=\sum a_nx^n$ 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 $b_n=-a_0^{-1}\sum_{i=1}^na_ib_{n-i}$ 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 $A/\mathfrak m$.
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