This one is related: Generalized Chinese Remainder Theorem
closed as not a real question by Victor Protsak, Andrew Stacey, Theo Johnson-Freyd, Bill Johnson, S. Carnahan♦ Sep 10 '10 at 0:29
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There is a chapter in the award-winning book The Mathematical Experience by Philip J. Davis and Reuben Hersh which can serve as an answer to this question. Its title is
"The Drive to Generality and Abstraction. The Chinese Remainder Theorem: A Case Study"
See (if you can)
It is an interesting take in part because Davis and Hersh are not algebraists or number theorists. Note that they do not include the result on pairwise comaximal ideals in a commutative ring, which is the most general result that I have in mind when I say "Chinese Remainder Theorem". On the other hand, they do discuss the interpretation of weak approximation as a sort of CRT. (In my view this is a good analogy but not precisely right: when both results apply, CRT has a stronger conclusion than weak approximation.)
This MO question contains a nice version for groups and half a Chinese remainder theorem (at most) for general algebraic structures.