In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule.
Definition
If and are rings and is a -bimodule, then the triangular matrix ring consists of 2-by-2 matrices of the form , where and with ordinary matrix addition and matrix multiplication as its operations.
![{\displaystyle R:=\left[{\begin{array}{cc}T&0\\M&U\\\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/572b4b468de48c98ac2093e9ed64b7b26159cb1a)
![{\displaystyle \left[{\begin{array}{cc}t&0\\m&u\\\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1ec750ef8b36dea5310dcc28844f5d735d0b0c)
References
- 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