criterion for cyclic rings to be principal ideal rings


A cyclic ring is a principal ideal ring if and only if it has a multiplicative identityPlanetmathPlanetmath.


Let R be a cyclic ring. If R has a multiplicative identity u, then u generates ( the additive groupMathworldPlanetmath of R. Let I be an ideal of R. Since {0R} is principal, it may be assumed that I contains a nonzero element. Let n be the smallest natural numberMathworldPlanetmath such that nuI. The inclusion nuI is trivial. Let tI. Since tR, there exists a with t=au. By the division algorithmPlanetmathPlanetmath, there exists q,r with 0r<n such that a=qn+r. Thus, t=au=(qn+r)u=(qn)u+ru=q(nu)+ru. Since ru=t-q(nu)I, by choice of n, it must be the case that r=0. Thus, t=q(nu). Hence, nu=I, and R is a principal ideal ring.

Conversely, if R is a principal ideal ring, then R is a principal idealMathworldPlanetmathPlanetmath. Let k be the behavior of R and r be a generator ( of the additive group of R such that r2=kr. Since R is principal, there exists sR such that s=R. Let a such that s=ar. Since rR=s, there exists tR with st=r. Let b such that t=br. Then r=st=(ar)(br)=(ab)r2=(ab)(kr)=(abk)r. If R is infiniteMathworldPlanetmathPlanetmath, then abk=1, in which case k=1 since k is nonnegative. If R is finite, then abk1mod|R|, in which case k=1 since k is a positive divisorMathworldPlanetmathPlanetmath of |R|. In either case, R has behavior one, and it follows that R has a multiplicative identity. ∎

Title criterion for cyclic rings to be principal ideal rings
Canonical name CriterionForCyclicRingsToBePrincipalIdealRings
Date of creation 2013-03-22 15:57:03
Last modified on 2013-03-22 15:57:03
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 15
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 16U99
Classification msc 13A99
Classification msc 13F10
Related topic CyclicRing3
Related topic PrincipalIdealRing
Related topic MultiplicativeIdentityOfACyclicRingMustBeAGenerator
Related topic CyclicRingsOfBehaviorOne