Triangular matrix ring

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


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.