cyclic rings that are isomorphic to $k\mathbb{Z}$

Corollary.

An infinite cyclic ring (http://planetmath.org/CyclicRing3) with positive behavior $k$ is isomorphic to $k\mathit{}\mathrm{Z}$.

Proof.

Note that $k\mathbb{Z}$ is an and that $k$ is a generator (http://planetmath.org/Generator) of its additive group. Since $k^2=k(k)$, then $k\mathbb{Z}$ has behavior $k$.∎