# Jacobson radical of a direct product of matrix rings

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?

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So, what do you know about finding the radical of a ring? – Gerry Myerson Jan 26 '13 at 5:25
I know formal definition that state: "The Jacobson radical J(R) of a unital ring R is the intersection of the annihilators of simple left R -modules." – rese Jan 26 '13 at 5:31
For commutative rings what you said can be reformulated as follows: the Jacobson radical is the intersection of all maximal ideals. – user26857 Feb 1 '13 at 18:25

It's true that $J(A\times B)=J(A)\times J(B)$. Then use that $J(\mathbb Z)=(0)$, $J(\mathbb Z_8)=\hat 2\mathbb Z_8$, and $J(\mathbb Z_3)=(\hat 0)$.