有限个代数元的质环

有限个代数元的质环 PRIME RINGS WITH A FINITE NUMBER OF ALGEBRAIC ELEMENTS 邱琦章 Qiu Qizhang 1 first-author 武汉大学数学系 武汉大学数学系 Department of Mathematics, Wuhan University Department of Mathematics, Wuhan University 设R是质环,具有有限个异于零的代数元。本文证明:若R有零因子,则R是有限域上的全矩阵环;若R无零因子,则R的所有代数元作成R的有限子域。 Let R be a ring. x∈R is called a algebraic element, if x m +α 1 x m-1 +……+a m-1 x=0 where α i ∈Z, i=1,2,……,m-1. Theoreml. Supposal R be a prime ring with a finite number of algebraic elements （≥2）. （1）If R possesses zero divisors, then RM n （F）, the total matrix ring over field F of degree n, where n≥2, F=GF（p m ） （2）If R possesses no zero divisors, then （ⅰ） The set of all algebraic elements of R is a finite field GF（p n ） （ⅱ） R is the direct sum of the left GF（p n ） spaces R 0 , R 1 ,……,R n 0 , where n 0 |n, R i ={x∈R|xa=a ( p i（n/n 0 ） x, 质环 有限域 全矩阵环 代数元 正则元 1992-01-01 2021-04-01 1