Does there exist a non-unital commutative ring such that all its proper ideals are prime?
Note also that that equipping the abelian group $\mathbb Z/p\mathbb Z$ with trivial multiplication $xy=0$ for all $x,y$ does not work. Though the resulting ring does not contain any nontrivial proper ideals, it does contain the trivial ideal $(0)$ which is not prime since the ring in question is not integral.
This question is inspired by an exercise which asked one to prove that all nontrivial rings with no non-prime proper ideals are fields. I suspect though that the text of the exercise simply forgot to specify the rings having units or was operating under some quiet "all rings are unital" assumption, as then it is rather simple to prove.