Doubts: All ideals of $M_n(R)$ are of the form $M_n(I)$, where $I$ is an ideal of $R$. I know the result is true, however, if we are to consider $E_{ij} = \left\{\begin{pmatrix} 0 & 0 & ... & 0\\
. & . & ...& . \\
0 & 0 & r_{ij} & 0 \\
. & . & ...& . \\ \end{pmatrix} : r_{ij}\in R  \right\}$. 
$E_{ij}$ is an ideal of $M_n(R)$ but doesn't appear to be of the type $M_n(I)$.
Can someone please explain what I am doing wrong here?

Your set is not closed by multiplication on the right.
You can multiply a nonzero element like that by a matrix with a 1 in the $j,1$ position and zeros elsewhere, and that is not in your set.
