So, a reduced ring has no nonzero nilpotent elements, while integral domain has no nonzero zero divisors. Of course, the latter is way stronger then the first condition, implying that every integral domain needs to be reduced ring. But, having $(\exists n \geq 2) x^n = 0 \Longleftrightarrow x = 0$ does not up to my knowledge imply $xy=0 \Longleftrightarrow (x = 0 \vee y = 0)$.
That is why I need example of reduced ring which is not integral domain. Thanks!
Best Answer
$\Bbb Z[X,Y]/(XY)$ has no non-zero nilpotents, but $XY = 0$.