This will hopefully be my last question about the chinese remainder theorem. I have asked several questions in an attmept to get a general version without conditions on the ideals which will trivially imply the usual version if they are assumed comaximal.
The answer to this question prompts me to ask what is the equalizer of the natural arrows below, sending $(r_j+I_j)_j$ to $(r_j+I_j+I_i)_{i,j}$ and $(r_i+I_i+I_j)_{i,j}$: $$\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$$ The elegant example given there shows the equalizer is not generally $R/\bigcap_j I_j$, but I'm having trouble figuring out what it is. I'm familiar with only the most basic definitions about affine schemes.