From the formulation of your question, it sounds like you are interested in associative but not-necessarily-commutative rings.  I cannot say anything about those.  However, for every commutative Noetherian ring $R$, for all maximal Cohen-Macaulay modules $M$ and $N$, the natural $R$-module homomorphism, $$\beta_{R,M,N}:\text{Hom}_R(M,R)\otimes_R N \to \text{Hom}_R(M,N),$$ is an isomorphism.  First, since $\beta_{M,N}$ is a natural transformation of additive functors, this is obviously an isomorphism when $N$ is a finite free $R$-module.  Second, since this is a local problem, it suffices to prove the case when $R$ is a local, Noetherian, Gorenstein ring of some dimension $d$ (local Noetherian rings have finite Krull dimension).  The result is proved by induction on $d$.  I will use some results from Chapter 21 of the following. (<b>Aside:</b> that is the second time I have mentioned Chapter 21 in two days.)

MR1322960 (97a:13001) <br>
Eisenbud, David(1-BRND) <br>
Commutative algebra. With a view toward algebraic geometry. <br>
Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. <br> 
ISBN: 0-387-94268-8; 0-387-94269-6 <br>
13-01 (14A05) <br>

In the base case that $d=0$, by Proposition 21.11, both $M$ and $N$ are finite free $R$-modules.  Thus by the first observation above, $\beta_{R,M,N}$ is an isomorphism.  Thus, by way of induction, assume that $d>0$, and assume that the result has been proved for smaller values of $d$.  Since $R$ is Gorenstein, it is Cohen-Macaulay, so that $\text{depth}_R(M) = d > 0$.  So there exists a system of parameters $(x_1,\dots,x_d)$ for $R$.  By Proposition 21.9, this system of parameters is simultaneously an $M$-sequence and an $N$-sequence.  In particular, consider the $R/x_dR$-module homomorphism induced from $\beta_{R,M,N}$,
$$ \beta_{R,M,N}\otimes \text{Id} : \text{Hom}_R(M,R)\otimes_R N \otimes_R(R/x_d R) \to \text{Hom}_R(M,N)\otimes_R (R/x_d R).$$  By Proposition 21.13, to prove that $\beta_{R,M,N}$ is an isomorphism, it is equivalent to prove that $\beta_{R,M,N}\otimes\text{Id}$ is an isomorphism.  

Since $R$ is a local Gorenstein ring of dimension $d$ and $x_d\in \mathfrak{m}_R$ is a nonzerodivisor, the quotient ring $R/x_d R$ is a local, Noetherian, Gorenstein ring (there are other references, but this also follows from Exercise 21.20).  Also, since $(x_1,\dots,x_d)$ is both an $M$-sequence and an $N$-sequence, by definition, also the induced system of parameters $(\overline{x}_1,\dots,\overline{x}_{d-1})$ is both an $M/x_dM$-sequence and an $N/x_dN$-sequence.  Therefore, by Proposition 21.9 again, both $M/x_dM$ and $N/x_dN$ are maximal Cohen-Macaulay modules over $R/x_d R$.  Thus, by the  induction hypothesis, the $R/x_d R$-module homomorphism,$$ \beta_{R/x_dR,M/x_dM,N/x_dN}: \text{Hom}_{R/x_d R}(M/x_d M,R/x_d R)\otimes_{R/x_d R} N/x_dN \to \text{Hom}_{R/x_d R}(M/x_d M, N/x_d N),  $$ is an isomorphism.  Finally, since $R$, $M$ and $N$ are maximal Cohen-Macaulay $R$-modules, by Proposition 21.12(b) (applied twice), $\beta_{R/x_dR,M/x_dM,N/X_dN}$ is the same as $\beta_{R,M,N}\otimes \text{Id}$.  Therefore $\beta_{R,M,N}$ is an isomorphism.  By induction, the result holds for all $d$.