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Thread: Simple Ring

  1. #1
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    Simple Ring

    Note: A simple Ring is a Ring $\displaystyle A \neq {0}$ 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 $\displaystyle A \neq {0}$ 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 $\displaystyle A=M_n(F), \ n \geq 2,$ the ring of all $\displaystyle n \times n$ matrices with entries in a field $\displaystyle F$. it's obviously not a division ring because not every non-zero matrix is invertible.

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

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