关于$3\times 3$阶三角矩阵环上模的研究

Study of Modules over $3\times 3$ Formal Triangular Matrix Rings

中文关键词:  三角矩阵环  一致模  空的模  根  基座.

英文关键词:triangular matrix ring  uniform module  hollow module  radical  socle.

中文摘要:

      文章对$3\times 3$阶三角矩阵环$$\Gamma = \left(\begin{array}{ccc}T & 0 & 0 \\M & U & 0\\{N \otimes _U M} & N & V \\\end{array}\right)$$上的模作了研究,其中T,U,V均是环, M,N分别是U-T, V-U双模.通过用一个五元组$(A,B,C;f,g)$来描述一个左$\Gamma$-模 (其中$A \in \mod T, B\in {\rm mod} U, C \in {\rm mod} V$, $f:M \otimes _T A \to B \in {\rm mod} U, g:N \otimes _U B \to C \in {\rm mod} V$), 文章分别刻画了$\Gamma$上的一致模、空的模、有限嵌入模,并且确定了${ }_\Gamma (A \oplus B \oplus C)$的根和基座．

英文摘要:

      In this paper we carry out a study of modules over a $3\times 3$ formal triangular matrix ring \[\Gamma= \left(\begin{array}{ccc}T & 0 & 0\\M & U & 0\\N\otimes_{U}M & N & V\end{array}\right) ,\] where $T$, $U$, $V$ are rings, $M$, $N$ are $U$-$T$, $V$-$U$ bimodules, respectively. Using the alternative description of left$\Gamma$-module as quintuple $(A,B,C;f,g)$ with $A\in \mod T$, $B\in \mod U$ and $C\in \mod V$, $f:M\otimes_{T}A\rightarrow B\in \mod U$, $g: N\otimes_{U}B\rightarrow C\in \mod V$, we shall characterize uniform, hollow and finitely embedded modules over $\Gamma$, respectively. Also the radical as well as the socle of $_{\Gamma}(A\oplus B\oplus C)$ is determined.