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
David E. Dobbs1
, John O. Kiltinen2
and Bobby J. Orndorff3
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
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.