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.
Best Answer
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 $n\ge2$.