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 $(U,T)$-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, https://books.google.com/books?isbn=0521599237