Let given ring $A$ without zero divisors and subring $\mathbf{k}\subset Z(A)$. Let also given $M~-$ $A~-$ bimodule, such that $xm = mx, \forall x\in \mathbf{k}, m\in M$.
Is it true that if for $\mathbf{F}:= Frac(\mathbf{k})$, vector spaces $A\otimes_{\mathbf{k}}\mathbf{F}, M\otimes_{\mathbf{k}}\mathbf{F}$ are finite dimensional over $\mathbf{F}$, then $HH^i(A\otimes_{\mathbf{k}}\mathbf{F}, M\otimes_{\mathbf{k}}\mathbf{F}) \cong HH^i(A, M)\otimes_{\mathbf{k}}\mathbf{F}$?
Where $HH^i(A, -)$ means Hochschild cohomology.
