= Cyclic finite quotients of the p-adic augmentation filtration
{title2=$\mathbb F_p[\mathbb Z_p]/I^{p^r}\cong\mathbb F_p[T]/(T^{p^r})$}
In the ordinary <group algebra> of the additive <p-adic integers>, let $I$ be the <augmentation ideal>. The <kernel> of the map to the <group algebra> of $\mathbb Z_p/p^r\mathbb Z_p$ is generated by $[p^rh]-1=([h]-1)^{p^r}$, so it is contained in $I^{p^r}$. The quotient is the cyclic-group algebra $\mathbb F_p[T]/(T^{p^r})$, so its <augmentation ideal> has zero $p^r$th power and the reverse inclusion holds. Every finite degree of the <associated graded ring> can be read from such a quotient, giving $\operatorname{gr}_I\mathbb F_p[\mathbb Z_p]\cong\mathbb F_p[X]$. The completion is $\mathbb F_p[[T]]$, but completion is not needed for the graded calculation.
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