cyclic ring

A ring is a cyclic ring if its additive groupMathworldPlanetmath is cyclic.

Every cyclic ring is commutativePlanetmathPlanetmathPlanetmath under multiplication. For if R is a cyclic ring, r is a generatorPlanetmathPlanetmathPlanetmath ( of the additive group of R, and s,tR, then there exist a,b such that s=ar and t=br. As a result, st=(ar)(br)=(ab)r2=(ba)r2=(br)(ar)=ts. (Note the disguised use of the distributive property (

A result of the fundamental theorem of finite abelian groups ( is that every ring with squarefree order is a cyclic ring.

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

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 kkn. R is an cyclic ring that has no zero divisorsMathworldPlanetmath if and only if there exists a positive integer k such that R is isomorphic to k. (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 𝐌2x2():


Thus, any cyclic ring that has zero divisors is a zero ringMathworldPlanetmath.


  • 1 Buck, Warren. 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