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Does there exist a simple ring with a nonsimple subring?

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Of course you are right... and For an example in a noncommutative setting we can take the ring of quaternions (which is simple since it is a division ring) and it's subring only the elements with integer coefficients - Z[i,j,k] –  asafsh Feb 26 at 10:24
    
@asafsh Please write what you finally discovered in a solution, so as to remove this question from the unanswered list. –  rschwieb Feb 26 at 13:27

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Z⊂Q is such an example. For an example in a noncommutative setting we can take the ring of quaternions (which is simple since it is a division ring) and it's subring only the elements with integer coefficients - Z[i,j,k]

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