TITLE: Is there any example of a simple Abelian ring which is not domain?
QUESTION [8 upvotes]: A ring $R$ is called:

simple if it has no two-sided ideal;
a domain if it has no zero divisor;
abelian if each idempotent of $R$ is central.

Is there any example of a simple abelian ring which is not a domain?

REPLY [4 votes]: Carl Faith, in
Faith, C., Noetherian simple rings, Bull. Am. Math. Soc. 70, 730-731 (1964). ZBL0132.02301
asked whether there was a simple Noetherian (i.e., both left and right Noetherian) ring that had no nontrivial idempotents but was not a domain.
This was answered with an example in 
Zalesskij, A. E.; Neroslavskij, O. M., There exist simple Noetherian rings with zero divisors but without idempotents, Commun. Algebra 5, 231-244 (1977). ZBL0352.16011.
That paper is in Russian, and I don't have access to it, but 
Lorenz, Martin, $K_ 0$ of skew group rings and simple Noetherian rings without idempotents, J. Lond. Math. Soc., II. Ser. 32, 41-50 (1985). ZBL0573.16004
gives more examples.
Maybe without the Noetherian condition there are easier examples, but I haven't thought of any.