can tensor product of two free non zero module over commutative ring with unity be zero?

And can tensor product of two non zero vector spaces be zero space?


Let $U$ and $V$ be free modules over a nontrivial commutative ring $R$ and let $\alpha:U\to R$ and $\beta:V\to R$ be two $R$-linear maps which have $1$ in their image; such things are easily seen to exist using freeness. Then using the properties of tensor products you can show that there is a morphism of abeelian groups $f:U\otimes_RV\to R$ such that for each $u\in U$ and each $v\in V$ we have $f(u\otimes v)=\alpha(u)\beta(v)$.

Now the hypothesis on $\alpha$ and $\beta$ tells us there exist $u_0\in U$ and $v_0\in V$ with $\alpha(u_0)=1$ and $\beta(v_0)=1$, and then $f(u_0\otimes v_0)=1$. This implies, of course, that $u_0\otimes v_0$ is a nonzero element of $U\otimes_RV$.

  • $\begingroup$ If $U$ is a right $R$-module and $V$ is a left $R$-module, and then we allow $R$ to be a possibly non-commutative,, exactly the same argument works. $\endgroup$ – Mariano Suárez-Álvarez May 23 '16 at 4:37
  • $\begingroup$ So will tensor product equal to LC of $e_i \otimes e_j$ where $e_i$ basis of $U$ and $e_j$ basis of $V$ $\endgroup$ – Sushil May 23 '16 at 4:41
  • $\begingroup$ While that is true, that most certainly does not follow from what I wrote... $\endgroup$ – Mariano Suárez-Álvarez May 23 '16 at 4:42
  • $\begingroup$ Okay but can you please tell why that will be true? $\endgroup$ – Sushil May 23 '16 at 4:47
  • $\begingroup$ I suggest you ask another question. Comments to an answer is a very bad place to ask new questions! $\endgroup$ – Mariano Suárez-Álvarez May 23 '16 at 4:49

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