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.