Let $R$ be a not necessarily commutative ring, and denote by $\_R\mathrm{lp}\_R$ the category of $R$-bimodules, which are finitely generated projective as left modules, with morphism $R$-bimodule maps, denoted by $\_R\mathrm{Hom}\_R(M,N)$, for any two objects $M,N$.


i) Is $\_R\mathrm{lp}\_R$ a monoidal category? In other words is $M \otimes\_R N$ again projective as a left $R$-module?


ii) Denote by $\_R\mathrm{Hom}(M,R)$ the left $R$-module maps from $M$ to $R$. Is $M^\* := \_R\mathrm{Hom}(M,R)$, endowed with its usual bimodule structure, projective as a left $R$-module?


iii) If $M^\*$ is projective, then is it a dual for $M^\*$, that is, is $\_R\mathrm{lp}\_R$ a rigid monoidal category?



Monoidal categories from the projective modules of a ring
Yes to (i). 


It's easier to think about if you weaken/generalize. 


Suppose that $M$ is an $(R,S)$-bimodule and $N$ is a left $S$-module. If $N$ is projective then the $R$-module $M\otimes\_SN$ is a summand of a direct sum of copies of $M\otimes\_SS\cong M$, so that it is a projective module if $M$ is.


No to (ii).


$\_RHom(M,R)$ gets its left module structure module from the right module structure of $M$. You can have an example where $R$ is a domain and where $M$ as a right module is a torsion module and $\_RHom(M,R)$ as a left module is a torsion module. For example, let $R=k[X]$, $k$ a field, take $M$ to be $R$ with the usual left module structure but right module structure given by $f(X)g(X)=f(X)g(0)$. (Bad notation, but I think you get the idea.)


