Thread: 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

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 is a ring with unity whose addition operation is matrix addition and multiplication is matrix multiplication.

If is a matrix that is invertible then, the ideal containing as an element must contain all elements of (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 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).