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Let $B = (B, \nabla, \eta, \Delta, \epsilon )$ be a bialgebra over a commutative ring $k$. Let $M$ and $N$ be two left $B$-modules. Then the tensor product $M \otimes_k N$ becomes a left $B$-module with multiplication rule given by

$ B \otimes_k M \otimes_k N \xrightarrow{\Delta \otimes 1} B \otimes_k B \otimes_k M \otimes_k N \xrightarrow{1 \otimes \tau \otimes 1} B \otimes_k M \otimes_k B \otimes_k N \xrightarrow{\mu \otimes \mu} M \otimes_k N$

where the $\mu$'s are the $B$-module multipication on $M$ and $N$. I am trying to show that $B$-$\mathsf{Mod}$ forms a monoidal category with tensor product $- \otimes_k -$. We have a $k$-linear associator $ \alpha_{MNP} \colon (M \otimes_k N) \otimes_k P \xrightarrow{\sim} M \otimes_k (N \otimes_k P)$ and I think this is supposed to be the associator in $B$-$\mathsf{Mod}$, so I need to show it is $B$-linear, i.e. that $b [m \otimes (n \otimes p)] = \alpha_{MNP} (b[(m \otimes n) \otimes p])$ but I am not really getting anywhere with it. Does anyone have any advice or at least be able to point me to a souce where this is covered? I am currently using Pareigis's Advanced Algebra, but proving monoidal structure is left as an exercise.

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    @Aaron Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-11

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