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ideals in matrix algebras
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(Topic)
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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
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Let
be an ideal in
.
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.
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"ideals in matrix algebras" is owned by mathcam. [ full author list (2) | owner history (1) ]
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(view preamble)
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 1591 times total.
Classification:
| AMS MSC: | 15A30 (Linear and multilinear algebra; matrix theory :: Algebraic systems of matrices) |
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Pending Errata and Addenda
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