# Simple unital ring which is not a domain

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.

• DaRT search for simple nondomain rings. – rschwieb Nov 18 '18 at 22:58
• The second thing you said is not conversely to the first thing. The converse would be "are all domains simple rings." – rschwieb Nov 18 '18 at 23:00

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 $$n\ge2$$.