Augmentation ideal of the infinite cyclic group ($\ker (\epsilon :\mathbb{Z}[t,t^{-1}]\to \mathbb{Z})=(t-1)\mathbb{Z}[t,t^{-1}]$)

The group ring of an infinite cyclic group is the Laurent polynomial ring $R=\mathbb{Z}[t,t^{-1}]$, with augmentation $\epsilon (t)=1$. 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.