Over a commutative ring, the tensor product of two free modules is free

Let R be a commutative ring with 1.

  1. Prove that the tensor product of two free R-modules is free.
  2. Prove that the tensor product of two projective R-modules is projective.

Suppose A and B are free R-modules; we can assume (since R is commutative) that A = \bigoplus_I R and B = \bigoplus_J R for some nonempty index sets I and J. By this previous exercise, binary tensor products essentially commute with direct sums. Thus, A \otimes_R B = (\bigoplus_I R) \otimes (\bigoplus_J R) \cong_R \bigoplus_I (R \otimes_R (\bigoplus_J R)) \cong_R \bigoplus_I \bigoplus_J (R \otimes_R R) \cong_R \bigoplus_{I \times J} R. Thus A \otimes_R B is free.

Now suppose A and B are projective. Then there exist M and N so that A \oplus M and B \oplus N are free. By the previous argument, (A \oplus M) \otimes_R (B \oplus N) is free. Now (A \oplus M) \otimes_R (B \oplus N) \cong_R A \otimes_R (B \oplus N) \oplus M \otimes_R (B \oplus N) \cong_R (A \otimes_R B) \oplus (A \otimes_R N) \oplus (M \otimes_R (B \oplus N)) is free, so that A \otimes_R B is projective.

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