ideals in matrix algebras
Let be a ring with 1. Consider the ring of -matrices with entries taken from .
It will be shown that there exists a one-to-onecorrespondence between the (two-sided) ideals of and the(two-sided) ideals of .
For , let denote the -matrix having entry 1 at position and 0 in all other places. It can beeasily checked that
(1) |
Let be an ideal in .
Claim.
The set given by
is an ideal in , and.
Proof.
since . Now let and be matrices in , and beentries 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 showsthat 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.