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-one correspondence 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 be easily checked that
Let be an ideal in .
The set given by
is an ideal in , and .
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|
|Date of creation||2013-03-22 13:59:28|
|Last modified on||2013-03-22 13:59:28|
|Last modified by||mathcam (2727)|