cyclic rings of behavior one


A cyclic ring has a multiplicative identityPlanetmathPlanetmath if and only if it has behavior one.


For a proof that a cyclic ring with a multiplicative identity has behavior one, see this theoremMathworldPlanetmath (

Let R be a cyclic ring with behavior one. Let r be a generatorPlanetmathPlanetmathPlanetmath ( of the additive groupMathworldPlanetmath of R such that r2=r. Let sR. Then there exists aR with s=ar. Since rs=r(ar)=ar2=ar=s and multiplicationPlanetmathPlanetmath in cyclic rings is commutativePlanetmathPlanetmathPlanetmath, then r is a multiplicative identity. ∎

Title cyclic rings of behavior one
Canonical name CyclicRingsOfBehaviorOne
Date of creation 2013-03-22 16:03:10
Last modified on 2013-03-22 16:03:10
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 9
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 13A99
Classification msc 16U99
Classification msc 13F10
Related topic MultiplicativeIdentityOfACyclicRingMustBeAGenerator
Related topic CriterionForCyclicRingsToBePrincipalIdealRings