Let $R$ be a commutative ring, and let $I, J \trianglelefteq R$ be ideals of $R$. Assume $\{0\} \notin \{I, J\}$. I had concluded, naively, that then
$$ \neg (\forall (i,j) ∈ I \times J)(i·j = 0). $$
However, $R$ need not be an integral domain, so this is not immediately guaranteed. I really do want to claim this, and I've left one property umentioned: $R$ is connected (i.e. $X^2-X ∈ R[X]$ has exactly two roots). I've already shown this to be equivalent to $R$ being nonzero and not isomorphic to any product of nonzero rings: $R \ncong R_1 \times R_2$.
I first had the idea that if I assumed that all products $i · j$ were zero, that would lead me to the contradiction that a subring of $R$ would be isomorphic to the product of $I$ and $J$. Then I realised that $I$, $J$ are only subrings if they contain a multiplicative identity, i.e. if they equal $R$... -.-
Can I claim the thing I want? Or is it simply untrue? And if so: why?
The claim is wrong if the ring is not an integral domain. Indeed, there are nonzero $a,b \in R$ with null product. Take $I=aR$, $J=bR$.