$Tensor Product$

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$Tensor Product$

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.

Abstract Algebra

Ssvnormandysr1

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We must show that every element of $F\otimes M$ is an $F$-linear combination of elements of $\{1\otimes x \}$. But elements of $F\otimes M$ are of the form $\sum_{i\le k} a_i\otimes m_i$ for some $a_i \in F$ and $m_i \in M$. So it suffices to show that each $a\otimes m$ is an $F$-linear combination of elements of $\{1\otimes x \}$. But $X$ generates $M$ so we have $m=\sum_{i\le n} r_i x_i$ for some $r_i \in R$ and $x_i \in X$. Now $$ a\otimes m=a\otimes (\sum r_i x_i )= \sum a\otimes r_i x_i =\sum ar_i \otimes x_i =\sum ar_i (1\otimes x_i )$$Note that $ar_i \in F$. So each $a\otimes m$ is an $F$-linear combination of elements of $\{1\otimes x \}$, and thus $$ \sum_{i\le k} a_i\otimes m_i=\sum_{i\le k}\sum _{j\le n_i} a_{ij} (1\otimes x_{ij})$$is also an $F$-linear combination of elements of $\{1\otimes x \}$.

Martin

Ssvnormandysr1

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

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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

$Tensor Product$

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