The direct sum of two modules is injective if and only if each direct summand is injective

Let R be a ring with 1 and let Q_1 and Q_2 be (left, unital) R-modules. Prove that Q_1 \oplus Q_2 is injective if and only if Q_1 and Q_2 are injective.


Recall Baer’s Criterion: an R-module Q is injective if and only if for every left ideal I \subseteq R and every R-module homomorphism \varphi : I \rightarrow Q, there exists an R-module homomorphism \Phi : R \rightarrow Q such that \Phi|_I = \varphi.

Suppose Q_1 and Q_2 are injective. Let I \subseteq R be a left ideal and let \varphi : I \rightarrow Q_1 \oplus Q_2 be an R-module homomorphism. Letting \pi_1 and \pi_2 denote the first and second coordinate projections, \pi_1 \circ \varphi : I \rightarrow Q_1 and \pi_2 \circ \varphi : I \rightarrow Q_2 are R-module homomorphisms. By Baer’s Criterion, there exist R-module homomorphisms \Phi_1 : R \rightarrow Q_1 and \Phi_2 : R \rightarrow Q_2 such that \Phi_1|_I = \pi_1 \circ \varphi and \Phi_2|_I = \pi_2 \circ \varphi. Define \Phi : R \rightarrow Q_1 \oplus Q_2 by \Phi(r) = (\Phi_1(r), \Phi_2(r)). Certainly \Phi is an R-module homomorphism. Moreover, \Phi_I =  \varphi. Thus Q_1 \oplus Q_2 is injective.

Suppose Q_1 \oplus Q_2 is injective. Let I \subseteq R be a left ideal and let \varphi_1 : I \rightarrow Q_1 and \varphi_2 : I \rightarrow Q_2 be R-module homomorphisms. Define \varphi : I \rightarrow Q_1 \oplus Q_2 by \varphi(i) = (\varphi_1(i), \varphi_2(i)). Certainly \varphi is an R-module homomorphism. By Baer’s Criterion, there exists an R-module homomorphism \Phi : R \rightarrow Q_1 \oplus Q_2 extending \varphi. Now let \Phi_1 = \pi_1 \circ \Phi and \Phi_2 = \pi_2 \circ \Phi; certainly \Phi_1 : R \rightarrow Q_1 and \Phi_2 : R \rightarrow Q_2 are R-module homomorphisms, and if a \in I then (\Phi_1(a), \Phi_2(a)) = \Phi(a) = \varphi(a) = (\varphi_1(a), \varphi_2(a)). Thus Q_1 and Q_2 are injective.

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Comments

  • yahyaalmalki  On December 10, 2011 at 4:47 am

    Hey!

    Thanks for the proof.

    We can do it also using the extension property of injective modules. In that case the proof is independent from Zorn’s Lemma (AC).

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