Quasi-Armendariz Modules

DOI：10.3770/j.issn:1000-341X.2010.04.018

 作者 单位 张翠萍 西北师范大学数学系，甘肃 兰州 730070 陈建龙 东南大学数学系, 江苏\ 南京 210096

对于右$R$-模$N$,我们引入了拟-Armendariz模的概念,它是Armendariz模和拟-Armendariz环的共同推广,研究了它的性质.此外,我们证明了$N_R$是拟-Armendariz的当且仅当$M_m(N)_{M_m(R)}$是拟-Armendariz的当且仅当$T_m(N)_{T_m(R)}$是拟-Armendariz的,其中$M_m(N)$和$T_m(N)$分别表示$N$上的$m\times m$全矩阵和$m\times m$上三角矩阵.$N_R$是拟-Armendariz的当且仅当$N[x]_{R[x]}$是拟-Armendariz的.每个拟-Baer模是拟-Armendariz的.

For a right $R$-module $N$, we introduce the quasi-Armendariz modules which are a common generalization of the Armendariz modules and the quasi-Armendariz rings, and investigate their properties. Moreover, we prove that $N_R$ is quasi-Armendariz if and only if $M_m(N)_{M_m(R)}$ is quasi-Armendariz if and only if $T_m(N)_{T_m(R)}$ is quasi-Armendariz, where $M_m(N)$ and $T_m(N)$ denote the $m\times m$ full matrix and the $m\times m$ upper triangular matrix over $N$, respectively. $N_R$ is quasi-Armendariz if and only if $N[x]_{R[x]}$ is quasi-Armendariz. It is shown that every quasi-Baer module is quasi-Armendariz module.