= Augmentation ideal of the infinite cyclic group
{title2=$\ker(\varepsilon:\mathbb Z[t,t^{-1}]\to\mathbb Z)=(t-1)\mathbb Z[t,t^{-1}]$}
The <group ring> of an <infinite cyclic group> is the Laurent polynomial ring $R=\mathbb Z[t,t^{-1}]$, with augmentation $\varepsilon(t)=1$. Multiply any element in the kernel by a power of $t$ to obtain a polynomial vanishing at one. The <factor theorem> makes it divisible by $t-1$. Thus the <augmentation ideal> is $(t-1)R$. Since $R$ is an <integral domain>, multiplication by $t-1$ is an injective module map, so this ideal is a free $R$-module of rank one, even though its additive group has infinite rank.
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