Kothe问题的等价条件

The equivalent conditions of Kothe problem

给出Kothe问题的一些等价条件.证明对任意环A,下列条件是等价的:1)设L是环A的任意诣零左理想,则L+LA是A的诣零理想;2)A的任意两个诣零左理想之和仍为A的诣零左理想;3)A的任意有限多个诣零左理想之和仍为A的诣零左理想;4)Nl(A)=U(A),这里U为Bear上诣零根;5)Ml(A)为A的诣零理想;6)对A的任意两个诣零左理想L1,L2,任意的r1,r2∈A,总有L1r1 +L2r2 是A的诣零左理想;7)对A的任意诣零左理想L,LA为A的诣零理想;8)对A的任意诣零左理想L及任意a∈A,L+La为A的诣零左理想.进而,通过证明Nl(A)=Nr(A)也获得了关于诣零左理想的等价条件,并讨论Kothe问题与Andrunakievich问题的关系.

Some equivalent conditions of the Kothe problem are given. It is shown that for every ring A, the following conditions are equevlent:1) Let L is any nil left ideal of A,then L+LA is a nil left ideal of A.2) Sum of any two nil left ideals of A is also nil left ideal of A. 3) Sum of any finitely many nil left ideals of A is also a nil left ideal of A. 4) N_l(A)=U(A),where U is nil radical of A. 5) M_l(A) is nil left ideal of A. 6) For any two nil left ideals L_1,L_2 of A,any r_1,r_2∈A, r_1L_1+r_2L_2 is also nil left ideal of A. 7) For any nil left ideal L of A, LA is nil left ideal of A. 8) For any nil left ideal L of A and a∈A,L+La is nil left ideal of A. Furthermore, some equivalent conditions related to nil right ideal of A are obtained by proving N_l(A)=N_r(A). Finally, the interrelation between Kothe problem and Andranaketvich problem is discussed.