cyclic rings of behavior one

Theorem.

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

Proof.

For a proof that a cyclic ring with a multiplicative identity has behavior one, see this theorem (http://planetmath.org/MultiplicativeIdentityOfACyclicRingMustBeAGenerator).

Let $R$ be a cyclic ring with behavior one. Let $r$ be a generator (http://planetmath.org/Generator) of the additive group of $R$ such that $r^{2}=r$. Let $s\in R$. Then there exists $a\in R$ with $s=ar$. Since $rs=r(ar)=ar^{2}=ar=s$ and multiplication in cyclic rings is commutative, then $r$ is a multiplicative identity. ∎

Title cyclic rings of behavior one CyclicRingsOfBehaviorOne 2013-03-22 16:03:10 2013-03-22 16:03:10 Wkbj79 (1863) Wkbj79 (1863) 9 Wkbj79 (1863) Theorem msc 13A99 msc 16U99 msc 13F10 MultiplicativeIdentityOfACyclicRingMustBeAGenerator CriterionForCyclicRingsToBePrincipalIdealRings