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:

1.

$R$ is a zero ring;
2.

$R$ has behavior $n$ ;
3.

$R\cong n\mathbb{Z}_{n^{2}}$ .

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 $k\equiv n\mod n$ . 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 $\mathbf{M}_{2\operatorname{x}2}(\mathbb{Z})$ :

$\mathbf{B}=\left\{\left.\left(\begin{array}[]{cc}n&-n\\ n&-n\end{array}\right)\right|n\in\mathbb{Z}\right\}.$

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 $\varphi\colon R\to\mathbf{B}$ defined by $\displaystyle\varphi(nr)=\left(\begin{array}[]{cc}n&-n\\ n&-n\end{array}\right)$ 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	2013-03-22 17:14:43
Last modified on	2013-03-22 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