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We define for rings with unity:

  • Simple: No non-trivial bilateral ideals.
  • Domain: No non-trivial zero divisors.

Are all simple rings domains? Conversely, are there any non-domain simple rings?

For commutative rings the question is trivial, as simple implies field. But I can't derive the existence of a non-trivial bilateral ideal from the existence of a non-trivial zero divisor.

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    $\begingroup$ DaRT search for simple nondomain rings. $\endgroup$ – rschwieb Nov 18 '18 at 22:58
  • $\begingroup$ The second thing you said is not conversely to the first thing. The converse would be "are all domains simple rings." $\endgroup$ – rschwieb Nov 18 '18 at 23:00
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The ring of $n\times n$ matrices over some field has just two bilateral ideals (zero and the whole ring), but it contains non-trivial zero divisors for all $n\ge2$.

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  • $\begingroup$ Thanks. Matrices were my first thought, but for some reason I assumed they wouldn't be simple. Shows what I know. $\endgroup$ – Emilio Martinez Nov 18 '18 at 19:11

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