Why is $M \otimes_R N$ not an $R$ module? My professor said in lecture that for $M$ a $(A, R)$ bimodule and $N$ a $(R, B)$ bimodule that $M \otimes_R N$ is not an $R$ module anymore but an abelian group and that it is naturally an $(A, B)$ bimodule. I can see how it is an $(A, B)$ bimodule but I think that it is also an $R$ module. Did I misunderstand my professor? We can still extend scalar multiplication. 

In general, $M\otimes N$ is not an $R$-module in any way at all —it is not that one has overlooked a way to do it.
For example, let $R=M_n(k)$, the matrix ring over a field of ome size $n>1$. Then the vector space $V$ of row vector of size $n$ is a right $R$-module in the obvious way, and the vector space $W$ of column vector is a left $R$-module, and $V\otimes_RW$ is not an $R$-module in any possible way.
Indeed, the vector space $V\otimes W$ is $1$-dimensional and if the field $k$ is finite then there is no $R$-module structure on it at all.
