诣零与幂零的若干经典问题

诣零与幂零的若干经典问题 SEVERAL CLASSICAL PROBLEMS OF NIL AND NILPOTENT 周柏荣 Zhou Borong 1 first-author 武汉大学数学系研究生班 学生杭州大学 武汉大学数学系研究生班 学生杭州大学 Mathematics Department Wuhan Universiy Mathematics Department Wuhan Universiy 本文利用相对极大左零化子（见定义）获得引理1、2,且给出一个环成立Kothe猜测、Herstein猜测、Kegel猜测的变形的简洁的充要条件。引理1 设R为环,X、Y为R的右理想且Y≤T。若R在M=Y-X上有关于Y的极大左零化子,则当M≠φ时必有R的右理想V≤Y,VX,而V 2 ≤X。引理2 设R为环,X为R的半质理想,Y为R的右理想且Y≤T,则Y≤X当且仅当R在Y-X上有关于Y的极大左零化子。 In this paper we get necessary and sufficient conditions on which a ring has Kothe conjecture, Herstein conjecture, transformation of Kegel conje-cture, ect. Let R be an associative ring, T=∑P α , for all of left nil p α △ L R, K be Kothe radical of R, L be Levitzki radical of R, N=∑N α , for all of ni-lpotent N α , △ R. Theorem 1 R has Kothe conjecture iff R has a maximal left amihilator relative to T in T-K. Theorem 3 R is local uilpotent iff R has a maximaal left annihilator relative to T in T-L. Theorem5 CN-ring R has Herstein conjectuse iff R has a maximal le-ft annihilator to T in T-K. 诣零 幂零 相对极大左零化子 nil nilpotent relative maximal left annihilator. 1988-03-01 2021-04-01 3