Tensor Product

Intro to Tensor Products exercise.

Let $R$ be an integral domain, $F$ its field of fractions and $M$ a left $R$-module.
Look at $F$ as an $F$-$R$-bimodule using the operations of $F$ so that $F?_RM$ has a structure of $F$-vector space.

  • Show that if $X$ is the generating set of $M$ as $R$-module, then the set $\{1?x|x?X\}$ generates $F?_RM$ as $F$-vector space. 

Answer

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  • Hey, are the a_i, m_i, r_i, x_i just arbitrary elements? Didn't quite figure that out.

  • yes they are arbitrary elements, although they may depend on the other elements introduced before. For example in the expansion of the arbitrary element m the r_i and x_i depend on m

The answer is accepted.