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

==References==

*{{Citation | last1=Auslander | first1=Maurice | last2=Reiten | first2=Idun | last3=Smalø | first3=Sverre O. | title=Representation theory of Artin algebras | orig-date=1995 | url=https://books.google.com/books?isbn=0521599237 | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-59923-8 | mr=1314422 | year=1997 | volume=36}}

Category:Ring theory