a ring $R$ is reduced if for each $a\in R$ , $a^n=0$ implies that $a=0$ for any positive integer $n$. also $R$ is called simple if it doesn't have any proper two-sided ideal.
Is there any example of a ring $R$ which is both simple and reduced and also not a domain?
Such rings do not exist. It was proved in 1968: Andrunakievič, V. A.; Rjabuhin, Ju. M. Rings without nilpotent elements, and completely prime ideals. Dokl. Akad. Nauk SSSR 180 1968 9–11.
From MathSci review;
...From here, it follows that a ring has no non-zero nilpotent elements if and only if it is isomorphic to a subdirect product of (non-commutative) integral domains. {This article has appeared in English translation [Soviet Math. Dokl. 9 (1968), 565–568].} Reviewed by V. Dlab