Bounds on Element Order in Rings Z(m) With Divisors of Zero

Authors

C. H. Cooke, Old Dominion University

Document Type

Article

Publication Date

2005

DOI

10.1016/j.camwa.2005.02.004

Publication Title

Computers & Mathematics with Applications

Volume

49

Issue

11-12

Pages

1643-1645

Abstract

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 = ₁p^αJ_J 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.

Comments

Web of Science: "Free full-text from publisher -- Elsevier open archive."

© 2005 Elsevier Ltd. All rights reserved.

Original Publication Citation

Cooke, C. H. (2005). Bounds on element order in rings Z_m with divisors of zero. Computers & Mathematics with Applications, 49(11-12), 1643-1645. doi:10.1016/j.camwa.2005.02.004

Repository Citation

Cooke, C. H., "Bounds on Element Order in Rings Z(m) With Divisors of Zero" (2005). Mathematics & Statistics Faculty Publications. 69.
https://digitalcommons.odu.edu/mathstat_fac_pubs/69