An arbitrary direct sum of modules is flat if and only if each direct summand is flat

Let R be a ring with 1 and let \{A_i\}_I be a family of (right, unital) R-modules. Prove that \bigoplus_I A_i is flat if and only if each A_i is flat.

We begin with a lemma.

Lemma: Let \{\varphi_i : A_i \rightarrow B_i\}_I be a family of (unital) R-module homomorphisms. Then \Phi : \bigoplus_I A_i \rightarrow \bigoplus_I B_i given by \Phi(a_i) = (\varphi_i(a_i)) is injective if and only if each \varphi_i is injective. Proof: Suppose \Phi is injective. If \varphi_k(a) = \varphi_k(b), and letting \iota_k denote the inclusion A_k \rightarrow \bigoplus_I A_i or B_k \rightarrow \bigoplus_I B_i, then \Phi(\iota_k(a)) = \Phi(\iota_k(b)), so that a = b. Thus each \varphi_k is injective. Conversely, suppose each \varphi_k is injective; then if \Phi(a_i) = \Phi(b_i), we have \varphi_i(a_i) = \varphi_i(b_i) for each i, so that a_i = b_i for each i. So (a_i) = (b_i), and thus \Phi is injective. \square

Suppose each A_i is flat. Now let L and M be left unital R-modules and let \theta : L \rightarrow M be a module homomorphism. Since each A_i is flat, 1_i \otimes \theta : A_i \otimes_R L \rightarrow A_i \otimes_R M is injective. Using the lemma, \Theta = \bigoplus_I (1_i \otimes \theta) : \bigoplus_I (A_i \otimes L) \rightarrow \bigoplus_I (A_i \otimes M) is injective. In this previous exercise, we constructed group isomorphisms \Phi : (\bigoplus_I A_i) \otimes_R L \rightarrow \bigoplus_I (A_i \otimes_R L) and \Psi : \bigoplus_I (A_i \otimes_R L) \rightarrow (\bigoplus_I A_i) \otimes_R M. We claim that \Psi \circ \Theta \circ \Phi = 1 \otimes \theta. To that end, note that (\Psi \circ \Theta \circ \Phi)((a_i) \otimes \ell) = (\Psi \circ \Theta)((a_i \otimes \ell)) = \Psi((a_i \otimes \theta(\ell))) = (\sum a_i) \otimes \theta(\ell) = (1 \otimes \theta)((a_i) \otimes \ell), as desired. Thus 1 \otimes \theta : (\bigoplus_I A_i) \otimes_R L \rightarrow (\bigoplus_I A_i) \otimes_R M is injective, and so \bigoplus_I A_i is flat.

Conversely, suppose \bigoplus_I A_i is flat. Let L and R be (unital, left) R-modules and \theta : L \rightarrow M a module homomorphism. Note that \Theta = 1 \otimes \theta : (\bigoplus_I A_i) \otimes_R L \rightarrow (\bigoplus_I A_i) \otimes_R M is injective. Recall the homomorphisms \Phi and \Psi from the previous paragraph; \Psi^{-1} \circ \Theta \circ \Phi^{-1} is an injective module homomorphism mapping \bigoplus_I (A_i \otimes_R L) to \bigoplus_I (A_i \otimes_R M), and in fact (\Psi^{-1} \circ \Theta \circ \Phi^{-1})((a_i \otimes \ell_i)) = (\Psi^{-1} \circ \Theta)(\sum \iota_i(a_i) \otimes \ell_i) = \Psi^{-1}(\sum \iota_i(a_i) \otimes \theta(\ell_i)) = (a_i \otimes \theta(\ell_i)) = ((1_i \otimes \theta)(a_i \otimes \ell_i)). By the lemma, each 1_i \otimes \theta is injective, so that each A_i is flat.

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