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 1≤i,j≤n, 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
Eij⋅Ekl={0iffk≠jEilotherwise. | (1) |
Let 𝔪 be an ideal in Mn×n(R).
Claim.
The set i⊆R given by
𝔦={x∈R∣x |
is an ideal in , and .
Proof.
since . Now let and be matrices in , and be entries of and respectively, say and . Then the matrix has at position , and it follows: If , then . Since is an ideal in it contains, in particular, the matrices and , where
thus, . This shows that is an ideal in . Furthermore, .
By construction, any matrix has entries in , so we have
so . Therefore . ∎
A consequence of this is: If is a field, then 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 |