On The Central Series Of A Kind Of Nilpotent Lie Rings

Let Q_?={n/m|(m,n)=1,n?Z,m is ? number},obviously Q_? is the subgroup of(Q,+).Let ?_ij be the prime set of(ij)positions in the matrix,then Q_?ij is Q_? corresponding to(ij)position.And let k_ij(1?i<j?n)be written as the product of prime powers,that is k_ij=p₁^e1p₂^e2…p_n^en.then there is each prime number Pi(?)?_ij.Let(?) then R is a subring of Lie ring if and only if the element at any position(ij)in the matrix satisfies the conditions that(?)and k_ij divides d_ij²,where d_ij²denotes the greatest common divisor of all k_irk_rj(1?i<r<j?n)and the Lie product of A,B is defined as[A,B]=AB-BA for arbitrary two elements A and B of R.Moreover,when R is a subring of Lie ring,both the upper central series and the lower central series of R coincide if and only if(?)and k_ij=d_ij^m(2?m?j-i).where d_ij^mdenotes the greatest common divisor of all k_il1k_l1l2…k_lm-1lj(1<l₁?l₂?…<l_m-1<j?n)and d_ij¹=k_ij if j-i=1.At last,this paper studies the structure of the subring of the nilpotent Lie ring,and discusses some cases where k_ij is zero,when the order of the matrix is 4.