Let $B$ be a local ring. Let $k$ be its residue field. Do we need $B$ to contain a copy of $k$ in order for the following to be true: $$\operatorname{Hom}_{k}({\Omega_{B/k}\otimes_{B} k},k)=\operatorname{Hom}_{B}(\Omega_{B/k},k)$$ as $k$-vector spaces, of course.
Thank you.