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tensor product is finitely generated

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Let $P$ be a $A$-$B$ bimodule which is finitely generated as a left $A$ module. Take $Q$ a $B$-$C$ bimodule that is finitely generated as a left $B$-module. Then show that $P\otimes_BQ$ is finitely generated as a left $A$-module.

I suppose the "base" of $P\otimes_BQ$ is the set $\{p_i\otimes q_i\}$ where the $p_i$ generate $P$ and the $q_i$ generate $Q$. However, I can't show that this is true.

$p\otimes q = \sum((a_ip_i)\otimes (b_jq_j))$ but there is no way of combining $q_j$ and $p_i$ to get an element in $A$.

I suspect that the exercise is not correctly stated.


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