Bounds on element order in rings Zm with divisors of zero

If p is a prime, integer ring Z_p has exactly ((p)) generating elements ω, each of which has maximal index I_p(ω) = (p) = p − 1. But, if m = Π^R_{J = 1} p^αJ_J is composite, it is possible that Z_m 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.