I want to show the map $\phi:M_\mathrm{n} (R) \rightarrow M_\mathrm{n} (R^{op})^{op}$ given by M$\rightarrow $$^t$M is an isomorphism of rings.
I have shown that it is injective and surjective but I am not sure how to show this is well defined and homomorphism .
If we take (R,+,.) and (R$^{op}$,+,*) then
$\phi$(M.N)=$^t$N * $^t$M but $\phi$(M) * $\phi$(N)= $^t$M * $^t$N