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 $I_1$, $I_2$, ..., $I_n$ be finitely many ideals of $R$ such that ($I_i+I_j=R$ for any two distinct elements $i$ and $j$ of $\left\lbrace 1,2,...,n\right\rbrace$). Then, $I_1I_2...I_n=I_1\cap I_2\cap ...\cap I_n$, and the canonical ring homomorphism $R/\left(I_1I_2...I_n\right)\to R/I_1 \times R/I_2 \times ... \times R/I_n$ 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 $I_1$, $I_2$, ..., $I_n$ be finitely many ideals of $R$ such that ($I_i+I_j=R$ for any two distinct elements $i$ and $j$ of $\left\lbrace 1,2,...,n\right\rbrace$). Let $A$ be an $R$-module. Then, $I_1I_2...I_n\cdot A=I_1A\cap I_2A\cap ...\cap I_nA$, and the canonical $R$-module homomorphism $A/\left(I_1I_2...I_n\cdot A\right)\to A/I_1A \times A/I_2A \times ... \times A/I_nA$ 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 $R\oplus A$ (with multiplication on $R\oplus 0$ inherited from $R$, multiplication between $R\oplus 0$ and $0\oplus A$ given by the $R$-module structure on $A$, and multiplication on $0\oplus A$ given by $0$), which seems quite artificial to me. Am I missing something very obvious?