Computers & Mathematics with Applications
If p is a prime, integer ring Zp has exactly ¢¢(p) generating elements ω, each of which has maximal index Ip(ω) = (p) = p − 1. But, if m = ΠRJ = 1pαJJ is composite, it is possible that Zm does not possess a generating element, and the maximal index of an element is not easily discernible. Here, it is determined when, in the absence of a generating element, one can still with confidence place bounds on the maximal index. Such a bound is usually less than ¢(m), and in some cases the bound is shown to be strict. Moreover, general information about existence or nonexistence of a generating element often can be predicted from the bound.
Original Publication Citation
Cooke, C. H. (2005). Bounds on element order in rings Zm with divisors of zero. Computers & Mathematics with Applications, 49(11-12), 1643-1645. doi:10.1016/j.camwa.2005.02.004
Cooke, C. H., "Bounds on Element Order in Rings Z(m) With Divisors of Zero" (2005). Mathematics & Statistics Faculty Publications. 69.