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[parent] behavior (Definition)

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.



"behavior" is owned by Wkbj79.
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behavior exists uniquely (infinite case) (Proof) by Wkbj79
behavior exists uniquely (finite case) (Proof) by Wkbj79
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Cross-references: divisor, positive, order, cyclic ring, finite, additive group, integer, infinite
There are 14 references to this entry.

This is version 12 of behavior, born on 2006-06-26, modified 2007-05-31.
Object id is 8091, canonical name is Behavior.
Accessed 1639 times total.

Classification:
AMS MSC: 13A99 (Commutative rings and algebras :: General commutative ring theory :: Miscellaneous)
 16U99 (Associative rings and algebras :: Conditions on elements :: Miscellaneous)
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behaviour by Wkbj79 on 2006-10-16 12:43:38
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". :-)
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nonstandard terminology by Wkbj79 on 2006-06-26 04:06:28
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.
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