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.

References

• Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995], Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, ISBN 978-0-521-59923-8, MR 1314422