The Central Series Of A Kind Of Nilpotent Lie Rings

Let R be a commutative ring with unity 1.And all the unit upper triangular matrices over the ring R whose order are n form a nilpotent group with respect to multiplication,denoted by U(n,R).This is the most classical example of nilpotent groups.Its upper and lower central series are coincident.Accordingly,the set of all upper triangular matrices of order n over R whose diagonal elements are 0 is denoted as L(n,R).L(n,R)is a nilpotent Lie ring with respect to the usual addition and Lie bracket and its upper and lower central series are coincident.In this paper,we will consider a nilpotent Lie ring which is more general than L(n,R).Let R be a commutative ring with 1,Jis a nilpotent ideal of R,satisfying Jm=0,Jm-1≠0.Then L is a nilpotent subring of full matrix ring of order n over R which is denoted as Mat(n,R).The nilpotent exponent of L is m+n-1.As a Lie ring,L is nilpotent.We calculate the nilpotent class and the upper and lower central series of L.The specific results are as follows:The upper central series of L is 0=ζ0L<ζ1L<…<ζm+n-2L=L,andζkL={(aij)∈L|aii=x∈J,aij∈Jm+n+i-j-k-1,i≠j},1≤k≤m+n-3.The lower central series of L is L=γ1L>γ2L>…>γm+n-1L=0,andγkL={(aij)∈L|aii=0,aij∈Ji-j+k,i≠j},2≤k≤m+n-1.We stipulate Jl denotes R,while l≤0.Furthermore,we study nilpotent Lie rings related to L.