This is a followup to Analog to the Chinese Remainder Theorem in groups other than Z_n. . I shouldn't have used the comments to ask a new question, in fact...
Here is the statement of the Chinese Remainder Theorem, as it occurs in most books and websites:
(1) Let R be a commutative ring with unity, and I1, I2, ..., In be finitely many ideals of R such that (Ii+Ij=R for any two distinct elements i and j of leftlbrace1,2,...,nrightrbrace). Then, I1I2...In=I1capI2cap...capIn, and the canonical ring homomorphism R/left(I1I2...Inright)toR/I1timesR/I2times...timesR/In is an isomorphism.
But there seems to be another, even more general form of (1) which doesn't get even half of the attention:
(2) Let R be a commutative ring with unity, and I1, I2, ..., In be finitely many ideals of R such that (Ii+Ij=R for any two distinct elements i and j of leftlbrace1,2,...,nrightrbrace). Let A be an R-module. Then, I1I2...IncdotA=I1AcapI2Acap...capInA, and the canonical R-module homomorphism A/left(I1I2...IncdotAright)toA/I1AtimesA/I2Atimes...timesA/InA is an isomorphism.
I am wondering: is (2) a trivial corollary of (1)? Because otherwise I don't see any reason why (2) shouldn't appear in literature as "the" Chinese Remainder Theorem, with (1) being but a corollary. Or is (2) wrong? The only way I see to get (2) from (1) is to apply (1) to the ring RoplusA (with multiplication on Roplus0 inherited from R, multiplication between Roplus0 and 0oplusA given by the R-module structure on A, and multiplication on 0oplusA given by 0), which seems quite artificial to me. Am I missing something very obvious?
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