# Help with ideal proof

• Oct 27th 2006, 07:35 AM
BenWong
Help with ideal proof
Yo. New to the board. Just looking for some help.

Let F be a field. Prove that the ring M2(F) of 2 x 2 matrices with entries in F has no non-trivial ideals. That is, every ideal is either the zero ideal or M2(F) itself.

Any help appreciated.
Benjamin
• Oct 27th 2006, 08:22 AM
ThePerfectHacker
Quote:

Originally Posted by BenWong
Yo. New to the board. Just looking for some help.

Let F be a field. Prove that the ring M2(F) of 2 x 2 matrices with entries in F has no non-trivial ideals. That is, every ideal is either the zero ideal or M2(F) itself.

Any help appreciated.
Benjamin

The proper terminology is a non-trivial proper ideal
Anyway,

The ring $M_2(F)$ is a ring with unity whose addition operation is matrix addition and multiplication is matrix multiplication.

If $A\in M_2(F)$ is a matrix that is invertible then, the ideal containing $A$ as an element must contain all elements of $M_2(F)$ (because any ideal containing a unit is improper) thus it is the ring itself.

Thus, the only way a non-trivial ideal can exist in $M_2(F)$ is when all the elements are non-units, that is non-invertible matrices. But that set cannot be an additive subgroup of the ring because addition would not be closed (since the sum of two non-invertible matrices can be invertible).