algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, a triangular matrix ring, also called a triangular ring, is a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

constructed from two rings and a

bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...

Definition

T

and

U

are rings and

M

is a

\left(U,T\right)

-bimodule, then the triangular matrix ring

R:=\left beginT&0\\M&U\\\end\right /math> consists of 2-by-2

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

of the form

\left begint&0\\m&u\\\end\right /math>, where t\in T,m\in M, and u\in U, with ordinary

matrix addition In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Kronec ...

and

matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

as its operations.

References

*{{Citation , last1=Auslander , first1=Maurice , last2=Reiten , first2=Idun , last3=Smalø , first3=Sverre O. , title=Representation theory of Artin algebras , origyear=1995 , url=https://books.google.com/books?isbn=0521599237 , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, series=Cambridge Studies in Advanced Mathematics , isbn=978-0-521-59923-8 , mr=1314422 , year=1997 , volume=36 Ring theory