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?
Best Answer
The second result you're talking about is also sometimes called the Chinese remainder theorem, and can be derived from the Chinese remainder theorem for rings by "tensoring the CRT isomorphism" with $A$. Explicitly, (1) gives
$R/\prod_{k=1}^n I_k\cong\prod_{k=1}^n R/I_k$
via the natural map. This is an isomorphism of rings as well as an isomorphism of $R$-modules. Therefore, upon tensoring with $A$, it becomes
$A/\big(\prod_{k=1}^nI_k\big)A\simeq \prod_{k=1}^n A/I_kA$
via the natural map, using the canonical isomorphism $R/I\otimes_RA\cong A/IA$ as well as the fact that tensor product commutes with finite direct products. It follows that the kernel of $A\rightarrow\prod_{k=1}^n A/I_kA$, which is clearly $\bigcap_{k=1}^n I_kA$, is equal to $\big(\prod_{k=1}^n I_k\big)A$. So, you've derived (2) from (1). Keep in mind that (2) is an isomorphism of $R$-modules, while (1) is an isomorphism of rings (as well as $R$-modules).