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Math Help - Help with ideal proof

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by BenWong View Post
    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).
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