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 $k$. ∎

Title	cyclic rings that are isomorphic to $k{\mathbb{Z}}_{kn}$
Canonical name	CyclicRingsThatAreIsomorphicToKmathbbZkn
Date of creation	2013-03-22 16:02:45
Last modified on	2013-03-22 16:02:45
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Corollary
Classification	msc 16U99
Classification	msc 13M05
Classification	msc 13A99
Related topic	MathbbZ_n