Let $R=M_5(\mathbb Z)\times M_3(\mathbb Z_8)\times M_3(\mathbb Z_3)$. How many elements has $J(R)$? (In this question $M_n(S)$ is referred to the ring of $n\times n$ matrices with elements in a ring $S$ and $J(R)$ is the Jacobson radical of ring $R$.)
I know that for any ring $R$ we have $$J(M_n(R))=M_n(J(R)).$$ For Jacobson radical and the direct product can we find an equality or inequality? For example the relation $J(A\times B)=J(A)\times J(B)$ can proved?