PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
ideals in matrix algebras (Topic)

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

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

Let $\mathfrak{m}$ be an ideal in $M_{n \times n}(R)$ .
Claim 1   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. $\mathfrak{i} \ne \emptyset$ 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 \begin{equation*} D_r :=\sum_{i=1}^n r\cdot E_{ii}, r \in R. \end{equation*}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 \begin{equation*} A=\sum\limits_{1 \le i,j \le n} a_{ij}E_{ij}, a_{ij} \in \mathfrak{i} \end{equation*}so $A \in m_{n \times n}(\mathfrak{i})$ . Therefore $\mathfrak{m} \subseteq M_{n \times n}(\mathfrak{i})$ . $ \qedsymbol$

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




"ideals in matrix algebras" is owned by mathcam. [ full author list (2) | owner history (1) ]
(view preamble | get metadata)

View style:

Log in to rate this entry.
(view current ratings)

Cross-references: simple, field, consequence, contains, matrices, places, ideals, one-to-one correspondence, ring

This is version 7 of ideals in matrix algebras, born on 2003-10-10, modified 2005-03-25.
Object id is 4767, canonical name is MatrixIdealsOverCommutativeRings.
Accessed 1938 times total.

Classification:
AMS MSC15A30 (Linear and multilinear algebra; matrix theory :: Algebraic systems of matrices)

Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)