# ideals in matrix algebras

Let $R$ be a ring with 1. Consider the ring $M_{n\times n}(R)$ of $n\times n$-matrices with entries taken from $R$.

It will be shown that there exists a one-to-one correspondence between the (two-sided) ideals of $R$ and the (two-sided) ideals of $M_{n\times n}(R)$.

For $1\leq i,j\leq n$, let $E_{ij}$ denote the $n\times n$-matrix having entry 1 at position $(i,j)$ and 0 in all other places. It can be easily checked that

 $E_{ij}\cdot E_{kl}=\left\{\begin{array}[]{lllll}0&\mbox{iff}&k\neq j\\ E_{il}&\mbox{otherwise.}\end{array}\right.$ (1)

Let $\mathfrak{m}$ be an ideal in $M_{n\times n}(R)$.

###### Claim.

The set $\mathfrak{i}\subseteq R$ given by

 $\mathfrak{i}=\{x\in R\mid x\quad\mbox{is an entry of }A\in\mathfrak{m}\}$

is an ideal in $R$, and $\mathfrak{m}=M_{n\times n}(\mathfrak{i})$.

###### Proof.

$\mathfrak{i}\neq\emptyset$ since $0\in\mathfrak{i}$. Now let $A=(a_{ij})$ and $B=(b_{ij})$ be matrices in $\mathfrak{m}$, and $x,y\in R$ be entries of $A$ and $B$ respectively, say $x=a_{ij}$ and $y=b_{kl}$. Then the matrix $A\cdot E_{jl}+E_{ik}\cdot B\in\mathfrak{m}$ has $x+y$ at position $(i,l)$, and it follows: If $x,y\in\mathfrak{i}$, then $x+y\in\mathfrak{i}$. Since $\mathfrak{i}$ is an ideal in $M_{n\times n}(R)$ it contains, in particular, the matrices $D_{r}\cdot A$ and $A\cdot D_{r}$, where

 $D_{r}:=\sum_{i=1}^{n}r\cdot E_{ii},r\in R.$

thus, $rx,xr\in\mathfrak{i}$. This shows that $\mathfrak{i}$ is an ideal in $R$. Furthermore, $M_{n\times n}(\mathfrak{i})\subseteq\mathfrak{m}$.

By construction, any matrix $A\in\mathfrak{m}$ has entries in $\mathfrak{i}$, so we have

 $A=\sum\limits_{1\leq i,j\leq n}a_{ij}E_{ij},a_{ij}\in\mathfrak{i}$

so $A\in m_{n\times n}(\mathfrak{i})$. Therefore $\mathfrak{m}\subseteq M_{n\times n}(\mathfrak{i})$. ∎

A consequence of this is: If $F$ is a field, then $M_{n\times n}(F)$ is simple.

Title ideals in matrix algebras IdealsInMatrixAlgebras 2013-03-22 13:59:28 2013-03-22 13:59:28 mathcam (2727) mathcam (2727) 10 mathcam (2727) Topic msc 15A30