矩阵环中的一类极大的广义的Armendariz子环

A Class of Maximal General Armendariz Subrings of Matrix Rings

中文关键词:  广义Armendariz环  矩阵环  广义reduced 环.

英文关键词:general Armendariz ring  matrix ring  general reduced ring.

中文摘要:

      如果在$R[x]$中,由$(\sum_{i=0}^{m}a_{i}x^{i})(\sum_{j=0}^{n}b_{j}x^{j})=0$,可得对于任意的$i$和$j$有 $a_{i}b_{j}=0$,那么称有单位元的结合环$R$为一个Armendariz环.一个有单位元的结合环$R$称为是reduced环, 若它没有非零的幂零元.在此定义了一个广义的reduced环(有或者没有单位元)和一个广义的Armendariz环(有或者没有单位元),并且给出了广义reduced环上的矩阵环中的一类极大的广义的Arm

英文摘要:

      An associative ring with identity $R$ is called Armendariz if, whenever $(\sum_{i=0}^{m}a_{i}x^{i})$ $(\sum_{j=0}^{n}b_{j}x^{j})=0$ in $R[x]$, $a_{i}b_{j}=0$ for all $i$ and $j$. An associative ring with identity is called reduced if it has no non-zero nilpotent elements. In this paper, we define a general reduced ring (with or without identity) and a general Armendariz ring (with or without identity), and identify a class of maximal general Armendariz subrings of matrix rings over general reduced rings.