It is known that $M\_n(R/J(R))\simeq M\_n(R)/M\_n(J(R))=M\_n(R)/J(M\_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R\_R)$, where $J(R)$ and $Soc(R\_R)$ stand for the Jacobson radical and the right socle of the ring $R$ (as a right $R$-module), but to no avail. Any help or suggestion would be welcome!



Isomorphism concerning $Soc(M_n(R))$
You can use the Morita equivalence between the categories $Mod(R)$ and $Mod(M\_n(R))$. The equiavlence is given by tensoring a right $R$ module with the bimodule $R^n$ over $R$ (in the other direction we also tensor with $R^n$, but over $M\_n(R)$). Under this equivalence, the $M\_n(R)$-module $M\_n(R)$ corresponds to the $R$-module $R^n$. Since being simple is a categorical property, and subobjects can also be defined categorically, we get that $Soc(R^n)=Soc(R)^n$, and this corresponds to the $M\_n(R)$-submodule $Soc(R)^n\otimes\_R R^n=Soc(R)^{n^2}$. In other words, we get that the socle of $M\_n(R)$ as an $M\_n(R)$-module is exactly the matrices whose entries are in $Soc(R)$.


