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

Question

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?

Answer 1

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.