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Let $R$ be a ring with identity, let $n\in \mathbb N^*$, and let $S=M_n(R)$ be the ring of $n\times n$ matrices with entries in $R$. Let $J$ be an ideal of $S$. Prove that the ideal $K := \{a \in R \mid a$ is the $(1,1)$ entry of some element of $J\}$ coincides with the set of elements of $R$ that occur as entries of elements of $J$.

First, I need to show that $M_n(K)=J$. I see it is kind of ambiguous to show that for any $A \in J$, $A \in M_n(k).$ Also, if $B \in M_n(K)$ it has entries in $K$, but how can we know that this entries are the same entries for a matrix in $A \in J$. I would apprecitate any help with that.

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