Monoidal categories from the projective modules of a ring

Question

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?

Answer 1

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.)