Matrix Rings

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 non-trivial 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 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 $R$ is a matrix ring. In fact, $R\cong M_n(S)$ , where $S$ is the centralizer of $T$ .
• 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 $n$ .

Matrix Groups