Recall, that an an associative algebra with unit $A$ is prime if whenever $a,b\in A$ have the property that for all $r\in A$, $arb=0$ then either $a=0$ or $b=0$. It is the correct way to extend the notion of integral domain to a noncommutative algebra, as matrix algebras have zero divisors, but they are prime.
Suppose now that $A$ and $B$ are prime algebras, then we can form the algebra $A\otimes B$ by letting $(a_1\otimes b_1)(a_2\otimes b_2)=a_1a_2\otimes b_1b_2$ and extending distributively.
Is $A\otimes B$ a prime algebra?
It is already not true that a tensor product of fields is integral: $\mathbb{C}\otimes_{\mathbb{R}} \mathbb{C}\simeq \mathbb{C}^2$.
(If you want a tensor product over $\mathbb{Z}$, you can adapt that example.)