环上矩阵的性质

环上矩阵的性质

摘要

设 是具有单位元的结合环， 为 上的矩阵环．本论文首先根据矩阵环的理想和剩余类环的定义，规定1个加法和乘法，构造出矩阵环的剩余类环．其次研究了环上初等矩阵的1些性质，并论证了初等矩阵群为1般线性群的正规子群，在此基础上研究环的 群， 环及其 群的性质.

关键词：矩阵环；剩余类环；1般线性群；特殊线性群；初等矩阵群

The Properties of Matrix Over Ring

ABSTRACT

Let be a associative ring with identity and be the matrix over . In this paper, Firstly, according to ideal of matrix ring and defines of residue class ring, ruled an addition and an multiplication, constructed a residue class ring of matrix ring. Secondly, We mainly invertigated the properties of elementary matrix group and proven the elementary group is normal subgroup in the general linear group and according to such properties, We discussed the properties of — group over and —ring over .

Keyword:matrix ring ; residue class ring ; general linear group ; special linear group ; elementary group