# cyclic ring

Every cyclic ring is commutative   under multiplication. For if $R$ is a cyclic ring, $r$ is a generator   (http://planetmath.org/Generator) of the additive group of $R$, and $s,t\in R$, then there exist $a,b\in{\mathbb{Z}}$ such that $s=ar$ and $t=br$. As a result, $st=(ar)(br)=(ab)r^{2}=(ba)r^{2}=(br)(ar)=ts.$ (Note the disguised use of the distributive property (http://planetmath.org/Distributive).)

A result of the fundamental theorem of finite abelian groups (http://planetmath.org/FundamentalTheoremOfFinitelyGeneratedAbelianGroups) is that every ring with squarefree order is a cyclic ring.

If $n$ is a positive integer, then, up to isomorphism       , there are exactly $\tau(n)$ cyclic rings of order $n$, where $\tau$ refers to the tau function. Also, if a cyclic ring has order $n$, then it has exactly $\tau(n)$ subrings. This result mainly follows from Lagrange’s theorem and its converse  . Note that the converse of Lagrange’s theorem does not hold in general, but it does hold for finite cyclic groups  .

Every subring of a cyclic ring is a cyclic ring. Moreover, every subring of a cyclic ring is an ideal.

$R$ is a finite cyclic ring of order $n$ if and only if there exists a positive divisor $k$ of $n$ such that $R$ is isomorphic to $k{\mathbb{Z}}_{kn}$. $R$ is an cyclic ring that has no zero divisors  if and only if there exists a positive integer $k$ such that $R$ is isomorphic to $k{\mathbb{Z}}$. (See behavior and its attachments for details.) Finally, $R$ is an cyclic ring that has zero divisors if and only if it is isomorphic to the following subset of ${\mathbf{M}}_{2\operatorname{x}2}({\mathbb{Z}})$:

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

## References

• 1 Buck, Warren. http://planetmath.org/?op=getobj&from=papers&id=336Cyclic Rings. Charleston, IL: Eastern Illinois University, 2004.
• 2 Kruse, Robert L. and Price, David T. Nilpotent Rings. New York: Gordon and Breach, 1969.
• 3 Maurer, I. Gy. and Vincze, J. “Despre Inele Ciclice.” Studia Universitatis Babeş-Bolyai. Series Mathematica-Physica, vol. 9 #1. Cluj, Romania: Universitatea Babeş-Bolyai, 1964, pp. 25-27.
• 4 Peinado, Rolando E. “On Finite Rings.” Mathematics Magazine, vol. 40 #2. Buffalo: The Mathematical Association of America, 1967, pp. 83-85.
Title cyclic ring CyclicRing 2013-03-22 13:30:13 2013-03-22 13:30:13 Wkbj79 (1863) Wkbj79 (1863) 33 Wkbj79 (1863) Definition msc 13A99 msc 16U99 CyclicGroup ProofOfTheConverseOfLagrangesTheoremForCyclicGroups CriterionForCyclicRingsToBePrincipalIdealRings MultiplicativeIdentityOfACyclicRingMustBeAGenerator