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

Let R be a ring with 1 and let P_1 and P_2 be (left, unital) R-modules. Prove that P_1 \oplus P_2 is projective if and only if P_1 and P_2 are projective.


Recall that, as it happens, a module is projective if and only if it is a direct summand of a free module.

Suppose P_1 \oplus P_2 is projective. Then for some module Q, we have P_1 \oplus P_2 \oplus Q free. In particular, both P_1 and P_2 are direct summands of a free module, and are thus projective.

Conversely, suppose P_1 and P_2 are projective; then there exist modules Q_1 and Q_2 such that P_1 \oplus Q_1 and P_2 \oplus Q_2 are free. As we proved in this previous exercise, (P_1 \oplus Q_1) \oplus (P_2 \oplus Q_2) \cong_R (P_1 \oplus P_2) \oplus (Q_1 \oplus Q_2) is free. Thus P_1 \oplus P_2 is projective.

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