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?