ac.commutative algebra - Chinese Remainder Theorem for rings: why not for modules?

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 $leftlbrace 1,2,...,nrightrbrace$). Then, $I_1I_2...I_n=I_1cap I_2cap ...cap I_n$, and the canonical ring homomorphism $R/left(I_1I_2...I_nright)to R/I_1 times R/I_2 times ... times R/I_n$ 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 $leftlbrace 1,2,...,nrightrbrace$). Let $A$ be an $R$-module. Then, $I_1I_2...I_ncdot A=I_1Acap I_2Acap ...cap I_nA$, and the canonical $R$-module homomorphism $A/left(I_1I_2...I_ncdot Aright)to A/I_1A times A/I_2A times ... times A/I_nA$ 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 $Roplus A$ (with multiplication on $Roplus 0$ inherited from $R$, multiplication between $Roplus 0$ and $0oplus A$ given by the $R$-module structure on $A$, and multiplication on $0oplus A$ given by $0$), which seems quite artificial to me. Am I missing something very obvious?