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 [math]\displaystyle{ T }[/math] and [math]\displaystyle{ U }[/math] are rings and [math]\displaystyle{ M }[/math] is a [math]\displaystyle{ \left(U,T\right) }[/math]-bimodule, then the triangular matrix ring [math]\displaystyle{ R:=\left[\begin{array}{cc}T&0\\M&U\\\end{array}\right] }[/math] consists of 2-by-2 matrices of the form [math]\displaystyle{ \left[\begin{array}{cc}t&0\\m&u\\\end{array}\right] }[/math], where [math]\displaystyle{ t\in T,m\in M, }[/math] and [math]\displaystyle{ u\in U, }[/math] 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