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 $$\left(U,T\right)$$-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.