In a commutative ring $R$ with unity, the product of every two comaximal ideals equals their intersection, that is, if $I + J = R$, then $I\cap J = IJ$.
The proof I know involves the unity of $R$, so that $1 = i + j$, and then $x = x\cdot 1= xi + xj = ix + xj \in IJ$.
Now, what if $R$ is a commutative rng, that is, without identity? Is it still true that $IJ = I\cap J$?