What is an example of a finitely generated $R$-bimodule $M$ satisfying

i) $M$ is projective as both a left and right $R$-module

ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual $_R\mathrm{Hom}(M,R)$ are isomorphic as bimodules,

where $R$ is a noncommutative unital algebra defined over a field $k$ with non-zero characteristic.

What is an example of a non-free finitely generated $R$-bimodule $M$ satisfying

i) $M$ is projective as both a left and right $R$-module

ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual $_R\mathrm{Hom}(M,R)$ are isomorphic as bimodules,

where $R$ is a noncommutative unital algebra defined over a field $k$ with non-zero characteristic.