Tensor product of free modules

Question

Suppose $M$ and $N$ are free $R$-module($R$ is a commutative ring). The tensor product of $M\otimes_R N$ is free $R$-module? I know for projective modules it is true. How should we build its basis?

A basis is the most natural possible: the tensor product of bases. — user26857, May 14 '17 at 19:11

score 23 · Accepted Answer · answered May 14 '17 at 19:11

23

$$[\bigoplus\limits_{i} R] \otimes_R [ \bigoplus\limits_j R] \cong \bigoplus\limits_{i,j} R \otimes_R R \cong \bigoplus\limits_{i,j} R$$

answered May 14 '17 at 19:11

D_S

35,843

That's right. And what about basis? – Bill May 14 '17 at 19:15
2

If you go through those isomorphisms explicitly, you should be able to see what the basis is – D_S May 14 '17 at 19:17

Tensor product of free modules

1 Answers1