子代数是李理想的结合代数

The Associative Algebras With Subalgebras Being Lie—Ideals

本文讨论子代数是李理想的结合代数,这种代数指的是域F上的一个结合代数A,若A的子代数都是A^-的理想。这是比H-代数更广泛的一类代数。若A是特征零域F上的这样的代数,我们得到以下主要结果:(1)设B,C,B+C及BC皆是A的子代数,若B或C诣零,则BC诣零;若B与C皆诣零,则B+C诣零;(2)若A诣零则A局部幂零;(3)若A是有限维的,则A/N=■(e_i),其中(e_i)是由e_i生成的A/N的理思,e_i^2=e_i(i=1,……,s)并且N是A的所有幂零元作成的A的幂零理想;(4)若A诣零,则对任意a∈A,a与ada有相同的幂零指数。

In[1],[2], Liu shaoxue has characterized the H—algebras. In this paper, we discuss a class of algebras more general than H—algebras. Assume A is an associative algebra over a field F. A is called the algebra with subalgebras being Lie—ideals, denoted by L-algebra, if its subalgebras are all the ideals of A^-. If F=0, A is a L-algtbra over F, we obtain the following main results: (1) Assume. B,C, B+Cand BC are subalgeras of A, if B is nil, then BC is nil; if B and C are nil, then B+C is nil also; (2) If A is nil then A is local nilpotent; (3) If A is finit dimensional, then A/N=(e_i), where(e_i), (j=1,…,s), are the ideals of A/N, generated by e_i, and N is the nilpolent ideal of A, which consist of all nilpotent elements of A; (4) If A is nil, then, for any element a∈A, a and ad a have the same nilpotent index.