Upper Triangular Matrix Lie Ring Over The Ring Of Integers

The group of unit upper triangular matrices over a ring is a very important example of a nilpotent group and has a very nice structure,where the upper and lower central series coincide.In general,its subsets do not form a group,and even in the case of a group,the regularity of its upper and lower central series becomes complicated.The Lie ring of upper triangular matrices over a ring is also a basic example of a nilpotent Lie ring,whose upper and lower central series are also congruent,and whose subsets do not form a Lie subring in general,and even in the case of a Lie ring,the structure of its upper and lower central series becomes complicated.In this paper,we will study the subrings of the ring of upper triangular matrices over the ring of integers,specifically,the set of any(i,j)-entry elements is some ideals of the ring of integers,denoted as(kij).We discuss the conditions to be the Lie rings,the conditions for the consistency of the upper and lower central series,the isomorphism problem with the associated Lie rings of the corresponding groups,etc.It is well known that the Lie ring method is known as a linearization method for study-ing nilpotent groups,which is a very important way of thinking for studying nilpotent groups.Therefore,the results of the nilpotent Lie rings studied in this paper are of interest to the study of nilpotent groups and to enrich the study of nilpotent Lie rings.Let n be a positive integer,2?r?n,we write(?) here kij(j-i? r-1,l?i<j?n)is a given positive integer.Let G=I+R={I+r| r?R},where I is the unit matrix of order n.This paper is divided into the following two parts.The first part proves that the sufficient condition for R to be a Lie ring is that kij divides dij(2)(j-r+1?i+r-1),where dij(12)is the greatest common divisor of all kijkij(i+r-1?l?j-r+1).When R is a Lie ring,the upper and lower central series of R are computed separately,and a sufficient condition for their upper and lower central series to coincide is further given.Then it is shown that when G is a group and the upper and lower central series of G coincide,its associated Lie ring L(G)is isomorphic to the Lie ring R,and the matrix representation of L(G)is obtained.In the second part,two special cases are discussed.First,let R*be the set of matrices in the Lie ring R where only the elements of the first row and the last column are non-zero,and give a sufficient condition for R*to be a Lie ring,and compute its upper and lower central series separately,and further give a sufficient condition for the upper and lower central series to coincide;second,discuss the subrings of the corresponding Lie ring R when k1r=0.Second,we discuss the subrings of the corresponding Lie ring R when k1r=0,and calculate their upper and lower central series respectively,and also give a sufficient condition for the coincidence of the upper and lower central series.