Lie环的两个幂零准则

Two nilpotency criteria for Lie rings

证明Lie环的两个幂零准则,即若Lie环L满足(ⅰ)L是可解的;(ⅱ)L/2γ(L)有限生成的;(ⅲ)对任意的x∈L,存在n∈N,使得x是左n-Engel的,则L是幂零的.且若条件(ⅱ)换成(ⅱ)′L满足中心化子上的极小条件,也可得L是幂零的.

Two nilpotency criteria for Lie rings are proved. Let L be a Lie ring and suppose that（ⅰ） L is soluble,（ⅱ）L/γ2（L） is finitely generated,（ⅲ）for each x∈L, there exists n∈N, such that x is left n-Engel, then L is niloptent. Moreover, if（ⅱ） is replaced by （ⅱ）′, L satisfies the minimal condition on centralizers,then L also is nilpotent.