cyclic rings and zero rings
Lemma 1.
Let $n$ be a positive integer and $R$ be a cyclic ring of order (http://planetmath.org/OrderRing) $R$. Then the following are equivalent^{}:
Proof.
To show that 1 implies 2, let $R$ have behavior $k$. Then there exists a generator^{} (http://planetmath.org/Generator) $r$ of the additive group^{} of $R$ such that ${r}^{2}=kr$. Since $R$ is a zero ring, ${r}^{2}={0}_{R}$. Since $kr={r}^{2}={0}_{R}=nr$, it must be the case that $k\equiv nmodn$. By definition of behavior, $k$ divides (http://planetmath.org/Divides) $n$. Hence, $k=n$.
The fact that 2 implies 3 follows immediately from the theorem that is stated and proven at cyclic rings that are isomorphic^{} to $k{\mathbb{Z}}_{kn}$ (http://planetmath.org/CyclicRingsThatAreIsomorphicToKmathbbZ_kn).
The fact that 3 implies 1 follows immediately since $n{\mathbb{Z}}_{{n}^{2}}$ is a zero ring. ∎
Lemma 2.
Let $R$ be an infinite^{} . Then the following are equivalent:

1.
$R$ is a zero ring;

2.
$R$ has behavior $0$;

3.
$R$ is isomorphic (http://planetmath.org/Isomorphism7) to the subring $\mathbf{B}$ of ${\mathbf{M}}_{2\mathrm{x}2}(\mathbb{Z})$:
$$\mathbf{B}=\left\{\left(\begin{array}{cc}\hfill n\hfill & \hfill n\hfill \\ \hfill n\hfill & \hfill n\hfill \end{array}\right)\rightn\in \mathbb{Z}\}.$$
Proof.
To show that 1 implies 2, the contrapositive of the theorem that is stated and proven at cyclic rings that are isomorphic to $k\mathbb{Z}$ (http://planetmath.org/CyclicRingsThatAreIsomorphicToKmathbbZ) can be used. If $R$ does not have behavior $0$, then its behavior $k$ must be positive by definition, in which case $R\cong k\mathbb{Z}$. It is clear that $k\mathbb{Z}$ is not a zero ring.
To show that 2 implies 3, let $r$ be a generator of the additive group of $R$. It can be easily verified that $\phi :R\to \mathbf{B}$ defined by $\phi (nr)=\left(\begin{array}{cc}\hfill n\hfill & \hfill n\hfill \\ \hfill n\hfill & \hfill n\hfill \end{array}\right)$ is a ring isomorphism.
The fact that 3 implies 1 follows immediately since $\mathbf{B}$ is a zero ring. ∎
Title  cyclic rings and zero rings 

Canonical name  CyclicRingsAndZeroRings 
Date of creation  20130322 17:14:43 
Last modified on  20130322 17:14:43 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  9 
Author  Wkbj79 (1863) 
Entry type  Result 
Classification  msc 13M05 
Classification  msc 13A99 
Classification  msc 16U99 