Triangular matrix ring

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

are rings and

-bimodule, then the triangular matrix ring

{\displaystyle R:=\left[{\begin{array}{cc}T&0\\M&U\\\end{array}}\right]}

consists of 2-by-2 matrices of the form

{\displaystyle \left[{\begin{array}{cc}t&0\\m&u\\\end{array}}\right]}

with ordinary matrix addition and matrix multiplication as its operations.