# 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 $k{\mathbb{Z}}_{kn}$.

###### 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 $k$. ∎

Title cyclic rings that are isomorphic to $k{\mathbb{Z}}_{kn}$ CyclicRingsThatAreIsomorphicToKmathbbZkn 2013-03-22 16:02:45 2013-03-22 16:02:45 Wkbj79 (1863) Wkbj79 (1863) 10 Wkbj79 (1863) Corollary msc 16U99 msc 13M05 msc 13A99 MathbbZ_n