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 $rs=r(ar)=a{r}^2=ar=s$ and multiplication in cyclic rings is commutative, then $r$ is a multiplicative identity.∎