Can the product of non-zero ideals in a unital ring be zero? Let $R$ be a ring with unity and $0\neq I,J\lhd R.$ Can it be that $IJ=0?$
It is possible in rings without unity. Let $A$ be a nontrivial abelian group made a ring by defining a zero multiplication on it. Then any subgroup of $S$ of $A$ is an ideal, because for any $a\in A,$ $s\in S,$ we have $$sa=as=0\in S.$$ Then if $S,T$ are nontrivial subgroups of $A,$ we have $ST=0.$
A non-trivial ring with unity cannot have zero multiplication, so this example doesn't work in this case. So perhaps there is no example? I believe there should be, but I can't find one. If there isn't one, is it possible in non-unital rings with non-zero multiplication?

Even simpler: Take any direct product of two nonzero rings $R\oplus S$. Then $R\oplus 0$ and $0\oplus S$ are nonzero ideals multiplying to zero.
One definition for prime ring is a ring in which this does not happen, so any non-prime ring will suffice.
Also, if you believe in nonzero nilpotent ideals, then you would easily find an example: If $I^n=\{0\}$ with $n$ minimal, then $I I^{n-1}=\{0\}$.
