# Monoidal categories from the projective modules of a ring

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?

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.
$$_RHom(M,R)$$ 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.)