criterion for cyclic rings to be principal ideal rings

Theorem.

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

Proof.

Let $R$ be a cyclic ring. If $R$ has a multiplicative identity $u$, then $u$ generates (http://planetmath.org/Generator) the additive group of $R$. Let $I$ be an ideal of $R$. Since $\{0_{R}\}$ is principal, it may be assumed that $I$ contains a nonzero element. Let $n$ be the smallest natural number such that $nu\in I$. The inclusion $\langle nu\rangle\subseteq I$ is trivial. Let $t\in I$. Since $t\in R$, there exists $a\in\mathbb{Z}$ with $t=au$. By the division algorithm, there exists $q,r\in\mathbb{Z}$ with $0\leq r such that $a=qn+r$. Thus, $t=au=(qn+r)u=(qn)u+ru=q(nu)+ru$. Since $ru=t-q(nu)\in I$, by choice of $n$, it must be the case that $r=0$. Thus, $t=q(nu)$. Hence, $\langle nu\rangle=I$, and $R$ is a principal ideal ring.

Conversely, if $R$ is a principal ideal ring, then $R$ is a principal ideal. Let $k$ be the behavior of $R$ and $r$ be a generator (http://planetmath.org/Generator) of the additive group of $R$ such that $r^{2}=kr$. Since $R$ is principal, there exists $s\in R$ such that $\langle s\rangle=R$. Let $a\in\mathbb{Z}$ such that $s=ar$. Since $r\in R=\langle s\rangle$, there exists $t\in R$ with $st=r$. Let $b\in\mathbb{Z}$ such that $t=br$. Then $r=st=(ar)(br)=(ab)r^{2}=(ab)(kr)=(abk)r$. If $R$ is infinite, then $abk=1$, in which case $k=1$ since $k$ is nonnegative. If $R$ is finite, then $abk\equiv 1\operatorname{mod}|R|$, in which case $k=1$ since $k$ is a positive divisor 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