Triangular matrix ring

In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule.

Definition

If $T$ and $U$ are rings and $M$ is a $(U,T)$-bimodule, then the triangular matrix ring $R:=\left[\begin{array}{cc}T& 0\\ M& U\end{array}\right]$ consists of 2-by-2 matrices of the form $\left[\begin{array}{cc}t& 0\\ m& u\end{array}\right]$, where $t\in T,m\in M,$ and $u\in U,$ with ordinary matrix addition and matrix multiplication as its operations.