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Math Help - Simple Ring

  1. #1
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    Simple Ring

    Note: A simple Ring is a Ring A \neq {0} such that it hasn't got proper ideals.

    I can prove that all division Ring is simple, but, is true that all simple ring is a division ring? If not, can you give a counter example?

    Thanks!
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  2. #2
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    Quote Originally Posted by roporte View Post
    Note: A simple Ring is a Ring A \neq {0} such that it hasn't got proper ideals.

    I can prove that all division Ring is simple, but, is true that all simple ring is a division ring? If not, can you give a counter example?

    Thanks!
    an easy example is A=M_n(F), \ n \geq 2, the ring of all n \times n matrices with entries in a field F. it's obviously not a division ring because not every non-zero matrix is invertible.

    the reason that A is simple is this well-known and easy to prove fact that for any ring R the two-sided ideals of M_n(R) are exactly in the form M_n(I), where I is a two-sided

    ideal of R. thus M_n(R) is simple if and only if R is simple.
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