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# 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.

Let $R$ be a cyclic ring with behavior one. Let $r$ be a 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. ∎

Related:

MultiplicativeIdentityOfACyclicRingMustBeAGenerator, CriterionForCyclicRingsToBePrincipalIdealRings

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

13A99*no label found*16U99

*no label found*13F10

*no label found*

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