Triangular matrix ring explained

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

Definition

and

are rings and

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\inT,m\inM,

and

u\inU,