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matrix ring

(Definition)

Matrix Rings

A ring

is said to be a matrix ring if there is a ring

and a positive integer

such that

$\displaystyle R\cong M_n(S),$

the ring of $n\times n$ matrices with entries as elements of

. Usually, we simply identify

with

Generally, one is interested to find out if a given ring is a matrix ring. By setting , we see that every ring is trivially a matrix ring. Therefore, to exclude these trivial cases, we call a ring a trivial matrix ring if there does not exist an such that $R\cong M_n(S)$ . Now the question becomes: is a non-trivial matrix ring?

Actually, the requirement that be a ring in the above definition is redundent. It is enough to define to be simply a set with two binary operations and $\cdot$ . Fix a positive integer $n\ge 1$ , define the set of formal $n\times n$ matrices with coefficients in . Addition and multiplication on are defined as the usual matrix addition and multiplication, induced by and $\cdot$ of respectively. By abuse of notation, we use and $\cdot$ to denote addition and multiplication on . We have the following:

If with is an abelian group, then so is .
If in addition, with both and $\cdot$ is a ring, then so is .
If is unital (has a multiplicative identity), then so is .

The first two assertions above are easily observed. To see how the last one roughly works, assume is the multiplicative identity of . Next define to be the matrix whose -cell is $a\in S$ and 0 everywhere else. Using cell entries $e_{st}$ from , we solve the system of equations

$\displaystyle U(e_{st},i,j)E=U(e_{st},i,j)=EU(e_{st},i,j)$

to conclude that

takes the form of a diagonal matrix whose diagonal entries are all the same element $e\in S$ . Furthermore, this

is an idempotent. From this, it is easy to derive that

is in fact a multiplicative identity of

(multiply an element of the form

, where

is an arbitrary element in

). The converse of all three assertions are clearly true too.

Remarks.

It can be shown that if is a unital ring having a finite doubly-indexed set $T=\lbrace e_{ij} \mid 1\le i,j\le n\rbrace$ such that
1. $e_{ij}e_{k\ell}=\delta_{jk}e_{i\ell}$ where $\delta_{jk}$ denotes the Kronecker delta, and
2. $\sum e_{ij}=1$ ,
then is a matrix ring. In fact, $R\cong M_n(S)$ , where is the centralizer of .
A unital matrix ring is isomorphic to the ring of endomorphisms of the free module . If has IBN, then $M_n(S)\cong M_m(S)$ implies that . It can also be shown that has IBN iff does.
Any ring is Morita equivalent to the matrix ring for any positive integer .

Matrix Groups

Suppose

is unital.

, the group of units of

, being isomorphic to the group of automorphisms of

, is called the general linear group of

. A matrix group is a subgroup of

for some matrix ring

. If

is a field, in particular, the field of real numbers or complex numbers, matrix groups are sometimes also called classical groups, as they were studied as far back as the 1800's under the name groups of tranformations, before the formal concept of a group was introduced.

"matrix ring" is owned by CWoo. [ full author list (2) ]

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Also defines:

matrix group

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Cross-references: classical groups, complex numbers, real numbers, field, subgroup, general linear group, automorphisms, group, group of units, Morita equivalent, iff, implies, IBN, free module, ring of endomorphisms, isomorphic, centralizer, Kronecker delta, finite, unital ring, converse, idempotent, diagonal, diagonal matrix, equations, cell, multiplicative identity, unital, abelian group, induced, matrix addition, multiplication, addition, coefficients, fix, binary operations, matrices, integer, positive, ring
There are 21 references to this entry.

This is version 8 of matrix ring, born on 2006-05-08, modified 2007-01-10.
Object id is 7908, canonical name is MatrixRing.
Accessed 3473 times total.

Classification:

AMS MSC:

16S50 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Endomorphism rings; matrix rings)

Pending Errata and Addenda

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