behavior
Behavior
2006-06-26 04:02:26
2007-05-31 02:27:44
Definition
cyclic ring
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If $R$ is an infinite \PMlinkname{cyclic ring}{CyclicRing3}, the {\sl behavior\/} of $R$ is a nonnegative integer $k$ such that there exists a \PMlinkname{generator}{Generator} $r$ of the additive group of $R$ with $r^2=kr$.
If $R$ is a finite cyclic ring of order $n$, the {\sl 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 \PMlinkescapetext{properties}.
To the best of my knowledge, this definition first appeared in my master's thesis:
Buck, Warren. \emph{\PMlinkexternal{Cyclic Rings}{http://planetmath.org/?op=getobj&from=papers&id=336}}. Charleston, IL: Eastern Illinois University, 2004.