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

Abstract:

      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.