Fermat小定理的一个注记

A note on Fermat's minor theorem

沿袭了文献 [1 ]的方法 ,将文献 [2 ,3 ]中的结论作了进一步推广 :设 Q是交换环 R的一个素理想 ,如果存在整数 n >1使 an≡ a(mod Q)对任何 a∈ R都成立 ,则 char(R/Q) =p,且 (1 )当|R/Q|≥ n时 ,存在 k∈ N使 n =pk;(2 )当 |R/Q|<n时 ,存在 r>0及 k≥ 0使 n =r(|R/Q|- 1 ) +pk。

The method of paper 1 is introduced in this article,and we further extend the solution to paper 2 and 3.Let Q be a prime ideal in commutation ring R,and n>1 be an integer,assume that for any a∈R,a n≡a(mod Q),we prove in this paper that char (R/Q)=p and n=p k for some integer k,unless R/Q is a finite field with |R/Q|<n.In the finite case,there exist k≥0 and r>0 such that n=r(|R/Q|-1)+p k.