Tensor Product of Projective Modules is Projective

Theorem
Let $A$ be a commutative ring with unity.

Let $P$ and $Q$ be projective $A$-modules.

Then the tensor product $P \otimes_A Q$ is a projective $A$-module.

Proof
By Projective iff Direct Summand of Free Module, there exist $A$-modules $P'$ and $Q'$, such that $P \oplus P'$ and $Q \oplus Q'$ are free.

By Tensor Product Distributes over Direct Sum, there is an isomorphism
 * $(P \oplus P') \otimes_A (Q \oplus Q') \cong (P \otimes_A Q) \oplus (P' \otimes_A Q) \oplus (P \otimes_A Q') \oplus (P' \otimes_A Q')$

By Tensor Product of Free Modules is Free the left hand side is free.

Hence $P \otimes_A Q$ is a direct summand of a free module.

By Projective iff Direct Summand of Free Module $P \otimes_A Q$ is projective.