International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 1, Pages 91-102
doi:10.1155/S0161171292000103
Commutative rings with homomorphic power functions
1Department of Mathematics, University of Tennessee, Knoxville 37996-1300, TN, USA
2Department of Mathematics & Comp. Sci., Northern Michigan University, Marquette 49855-5340, MI, USA
3Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061-0106, VA, USA
Received 25 July 1990; Revised 15 August 1991
Copyright © 1992 David E. Dobbs et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A (commutative) ring R (with identity) is called m-linear (for an integer m≥2) if (a+b)m=am+bm for all a and b in R. The m-linear reduced rings are characterized, with special attention to the finite case. A structure theorem reduces the study of m-linearity to the case of prime characteristic, for which the following result establishes an analogy with finite fields. For each prime p and integer m≥2 which is not a power of p, there exists an integer s≥m such that, for each ring R of characteristic p, R is m-linear if and only if rm=rps for each r in R. Additional results and examples are given.