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

Corollary.

A finite cyclic ring of order (http://planetmath.org/OrderRing) $n$ with behavior $k$ is isomorphic to $k\mathit{}{\mathrm{Z}}_{k\mathit{}n}$.

Proof.

Note that $k{\mathbb{Z}}_{kn}$ is a cyclic ring and that $k$ is a generator of its additive group. As groups, $k{\mathbb{Z}}_{kn}$ and ${\mathbb{Z}}_n$ are isomorphic. Thus, $k{\mathbb{Z}}_{kn}$ has order $n$. Since $k^2=k(k)$, then $k\mathbb{Z}$ has behavior $k$.∎