# behavior

If $R$ is an infinite cyclic ring (http://planetmath.org/CyclicRing3), the behavior of $R$ is a nonnegative integer $k$ such that there exists a generator (http://planetmath.org/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 .

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

Buck, Warren. http://planetmath.org/?op=getobj&from=papers&id=336Cyclic Rings. Charleston, IL: Eastern Illinois University, 2004.

Title behavior Behavior 2013-03-22 16:02:29 2013-03-22 16:02:29 Wkbj79 (1863) Wkbj79 (1863) 15 Wkbj79 (1863) Definition msc 13A99 msc 16U99