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
Canonical name	Behavior
Date of creation	2013-03-22 16:02:29
Last modified on	2013-03-22 16:02:29
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	15
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 13A99
Classification	msc 16U99