# 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 $1\le i,j\le n$, 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

$${E}_{ij}\cdot {E}_{kl}=\{\begin{array}{ccccc}0\hfill & \text{iff}\hfill & k\ne j\hfill & & \\ {E}_{il}\hfill & \text{otherwise.}\hfill & & & \end{array}$$ | (1) |

Let $\U0001d52a$ be an ideal in ${M}_{n\times n}(R)$.

###### Claim.

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

$$\U0001d526=\{x\in R\mid x\mathit{\hspace{1em}}\mathit{\text{is an entry of}}A\in \U0001d52a\}$$ |

is an ideal in $R$, and $\mathrm{m}\mathrm{=}{M}_{n\mathrm{\times}n}\mathit{}\mathrm{(}\mathrm{i}\mathrm{)}$.

###### Proof.

$\U0001d526\ne \mathrm{\varnothing}$ since $0\in \U0001d526$. Now let $A=({a}_{ij})$ and $B=({b}_{ij})$ be matrices in $\U0001d52a$, and $x,y\in R$ be entries of $A$ and $B$ respectively, say $x={a}_{ij}$ and $y={b}_{kl}$. Then the matrix $A\cdot {E}_{jl}+{E}_{ik}\cdot B\in \U0001d52a$ has $x+y$ at position $(i,l)$, and it follows: If $x,y\in \U0001d526$, then $x+y\in \U0001d526$. Since $\U0001d526$ 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, $rx,xr\in \U0001d526$. This shows that $\U0001d526$ is an ideal in $R$. Furthermore, ${M}_{n\times n}(\U0001d526)\subseteq \U0001d52a$.

By construction, any matrix $A\in \U0001d52a$ has entries in $\U0001d526$, so we have

$$A=\sum _{1\le i,j\le n}{a}_{ij}{E}_{ij},{a}_{ij}\in \U0001d526$$ |

so $A\in {m}_{n\times n}(\U0001d526)$. Therefore $\U0001d52a\subseteq {M}_{n\times n}(\U0001d526)$. ∎

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

Title | ideals in matrix algebras |
---|---|

Canonical name | IdealsInMatrixAlgebras |

Date of creation | 2013-03-22 13:59:28 |

Last modified on | 2013-03-22 13:59:28 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 10 |

Author | mathcam (2727) |

Entry type | Topic |

Classification | msc 15A30 |