Jacobson环可分解为域的直和的一个充分条件

A SUFFICIENT CONDITION OF THE JACOBSON RING SEPARATED INTO THE DIRECT SUM OF FIELDS

本文得到Jacobson环R在方程a_nx^n+…+a_1x=0(a_n、…、a_1∈Z(整数集)或环R,且■α∈R,a_nα+…+a_1α=0)上有有限个解的条件下,可分解为域的直和.由此给出,当上面的解的个数为素数时,则R 为域,从而推广了谢邦杰1982的结果.

In this paper,if the Jacobson ring R have fintie solutions on the equation a_nx^n+…+a_1x=0(a_n,…,a_1∈Z(the integerset)or the ring R,and (?) a∈R,a_na+…+a_1a=0),then R can be separted into the direct sum of fields.When the numberof the solutions is prime number,then the R is a field.so we generalize the resalt of Xie Bangjie in 1982.