cyclic ring
A ring is a cyclic ring if its additive group^{} is cyclic.
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 ${\mathbf{M}}_{2\mathrm{x}2}(\mathbb{Z})$:
$\left\{\left(\begin{array}{cc}\hfill c\hfill & \hfill -c\hfill \\ \hfill c\hfill & \hfill -c\hfill \end{array}\right)\right|c\in \mathbb{Z}\}$
Thus, any cyclic ring that has zero divisors is a zero ring^{}.
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 |
---|---|
Canonical name | CyclicRing |
Date of creation | 2013-03-22 13:30:13 |
Last modified on | 2013-03-22 13:30:13 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 33 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 13A99 |
Classification | msc 16U99 |
Related topic | CyclicGroup |
Related topic | ProofOfTheConverseOfLagrangesTheoremForCyclicGroups |
Related topic | CriterionForCyclicRingsToBePrincipalIdealRings |
Related topic | MultiplicativeIdentityOfACyclicRingMustBeAGenerator |