A Grothendieck ring of finite characteristic

We construct, for every integer $N\in\mathbb{N}^*$, a structure whose Grothendieck ring is isomorphic to $(\mathbb{Z}/N\mathbb{Z})[X]$, thus proving the existence of structures with a non-zero Grothendieck ring with non-zero characteristic. Namely, this structure consists of the bijection without cycles between a set and a complement of $N$ points in this set.