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. ∎

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