ideals in matrix algebras

Let R be a ring with 1. Consider the ring Mn×n(R) of n×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 Mn×n(R).

For 1i,jn, let Eij denote the n×n-matrix having entry 1 at position (i,j) and 0 in all other places. It can be easily checked that

EijEkl={0iffkjEilotherwise. (1)

Let 𝔪 be an ideal in Mn×n(R).


The set iR given by

𝔦={xRxis an entry of A𝔪}

is an ideal in R, and m=Mn×n(i).


𝔦 since 0𝔦. Now let A=(aij) and B=(bij) be matrices in 𝔪, and x,yR be entries of A and B respectively, say x=aij and y=bkl. Then the matrix AEjl+EikB𝔪 has x+y at position (i,l), and it follows: If x,y𝔦, then x+y𝔦. Since 𝔦 is an ideal in Mn×n(R) it contains, in particular, the matrices DrA and ADr, where


thus, rx,xr𝔦. This shows that 𝔦 is an ideal in R. Furthermore, Mn×n(𝔦)𝔪.

By construction, any matrix A𝔪 has entries in 𝔦, so we have


so Amn×n(𝔦). Therefore 𝔪Mn×n(𝔦). ∎

A consequence of this is: If F is a field, then Mn×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