This question has been asked on here before but I'm looking for some additional insights. The question comes from the section of the Dummit and Foote textbook on the Chinese Remainder Theorem. All of the proofs that I have seen so far don't use the CRT. Is there any way to prove this using the CRT?
I was thinking of trying to show that there exists a ring A with comaximal ideals I, J such that by the Chinese Remainder Theorem $R\times S \cong A/(IJ)$. Then because $A/(IJ)$ is not a field, $R\times S$ is not a field.