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# behavior

If $R$ is an infinite cyclic ring, the behavior of $R$ is a nonnegative integer $k$ such that there exists a generator $r$ of the additive group of $R$ with $r^{2}=kr$.

If $R$ is a finite cyclic ring of order $n$, the behavior of $R$ is a positive divisor $k$ of $n$ such that there exists a generator $r$ of the additive group of $R$ with $r^{2}=kr$.

For any cyclic ring, behavior exists uniquely. Moreover, the behavior of a cyclic ring determines many of its properties.

To the best of my knowledge, this definition first appeared in my master’s thesis:

Buck, Warren. *Cyclic Rings*. Charleston, IL: Eastern Illinois University, 2004.

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Reference

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Definition

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## Mathematics Subject Classification

13A99*no label found*16U99

*no label found*

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## Comments

## nonstandard terminology

As far as I am aware, the term "behavior" as I have defined it appeared for the first time in my master's thesis, Cyclic Rings. Also as far as I am aware, no one else has given this thing a name. Is it acceptable for me to define this on PM? It would make many of my proofs about cyclic rings much simpler.

## Re: nonstandard terminology

I would at least cite the reference (in this case your approved thesis) in the entry.

## behaviour

For the time being, I am removing "behaviour" from the synonyms section of this entry in an effort to reduce spurious links. I hope that does not offend anyone who spells "British style". :-)