In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule.
Definition
If T {\displaystyle T} and U {\displaystyle U} are rings and M {\displaystyle M} is a ( U , T ) {\displaystyle \left(U,T\right)} -bimodule, then the triangular matrix ring R := [ T 0 M U ] {\displaystyle R:=\left[{\begin{array}{cc}T&0\\M&U\\\end{array}}\right]} consists of 2-by-2 matrices of the form [ t 0 m u ] {\displaystyle \left[{\begin{array}{cc}t&0\\m&u\\\end{array}}\right]} , where t ∈ T , m ∈ M , {\displaystyle t\in T,m\in M,} and u ∈ U , {\displaystyle u\in U,} with ordinary matrix addition and matrix multiplication as its operations.
- Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995], Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, ISBN 978-0-521-59923-8, MR 1314422