• Source: 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


{\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.




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

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