# Matrices with ideals [duplicate]

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.