matrix ring
0.1 Matrix Rings
A ring $R$ is said to be a matrix ring if there is a ring $S$ and a positive integer $n$ such that
$$R\cong {M}_{n}(S),$$ 
the ring of $n\times n$ matrices with entries as elements of $S$. Usually, we simply identify $R$ with ${M}_{n}(S)$.
Generally, one is interested to find out if a given ring $R$ is a matrix ring. By setting $n=1$, we see that every ring is trivially a matrix ring. Therefore, to exclude these trivial cases, we call a ring $R$ a trivial matrix ring if there does not exist an $n>1$ such that $R\cong {M}_{n}(S)$. Now the question becomes: is $R$ a nontrivial matrix ring?
Actually, the requirement that $S$ be a ring in the above definition is redundent. It is enough to define $S$ 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 ${M}_{n}(S)$ with coefficients in $S$. Addition and multiplication on ${M}_{n}(S)$ are defined as the usual matrix addition^{} and multiplication, induced by $+$ and $\cdot $ of $S$ respectively. By abuse of notation, we use $+$ and $\cdot $ to denote addition and multiplication on ${M}_{n}(S)$. We have the following:

1.
If ${M}_{n}(S)$ with $+$ is an abelian group^{}, then so is $S$.

2.
If in addition, ${M}_{n}(S)$ with both $+$ and $\cdot $ is a ring, then so is $S$.

3.
If ${M}_{n}(S)$ is unital (has a multiplicative identity^{}), then so is $S$.
The first two assertions above are easily observed. To see how the last one roughly works, assume $E$ is the multiplicative identity of ${M}_{n}(S)$. Next define $U(a,i,j)$ to be the matrix whose $(i,j)$cell is $a\in S$ and $0$ everywhere else. Using cell entries ${e}_{st}$ from $E$, we solve the system of equations
$$U({e}_{st},i,j)E=U({e}_{st},i,j)=EU({e}_{st},i,j)$$ 
to conclude that $E$ takes the form of a diagonal matrix^{} whose diagonal entries are all the same element $e\in S$. Furthermore, this $e$ is an idempotent^{}. From this, it is easy to derive that $e$ is in fact a multiplicative identity of $S$ (multiply an element of the form $U(a,1,1)$, where $a$ is an arbitrary element in $S$). The converse of all three assertions are clearly true too.
Remarks.

•
It can be shown that if $R$ is a unital ring having a finite doublyindexed set $T=\{{e}_{ij}\mid 1\le i,j\le n\}$ such that

(a)
${e}_{ij}{e}_{k\mathrm{\ell}}={\delta}_{jk}{e}_{i\mathrm{\ell}}$ where ${\delta}_{jk}$ denotes the Kronecker delta^{}, and

(b)
$\sum {e}_{ij}=1$,
then $R$ is a matrix ring. In fact, $R\cong {M}_{n}(S)$, where $S$ is the centralizer^{} of $T$.

(a)

•
A unital matrix ring $R={M}_{n}(S)$ is isomorphic^{} to the ring of endomorphisms of the free module^{} ${S}^{n}$. If $S$ has IBN, then ${M}_{n}(S)\cong {M}_{m}(S)$ implies that $n=m$. It can also be shown that $S$ has IBN iff $R$ does.

•
Any ring $S$ is Morita equivalent to the matrix ring ${M}_{n}(S)$ for any positive integer $n$.
0.2 Matrix Groups
Suppose $R={M}_{n}(S)$ is unital. $U(R)$, the group of units of $R$, being isomorphic to the group of automorphisms^{} of ${S}^{n}$, is called the general linear group^{} of ${S}^{n}$. A matrix group is a subgroup^{} of $U(R)$ for some matrix ring $R$. If $S$ 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.
Title  matrix ring 

Canonical name  MatrixRing 
Date of creation  20130322 15:54:18 
Last modified on  20130322 15:54:18 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16S50 
Defines  matrix group 