Suppose that $R$ is a semisimple ring with unity. Let $S=M_n(R)$ be the matrix ring. For simplicity, we proceed with the special case $n=2$. Then, as easily seen, \begin{align} S=e_{11}S + e_{22}S, \end{align} where \begin{align} e_{11}= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad, e_{22}= \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}. \end{align} Hence, \begin{align} I:=e_{11}S= \left\lbrace \begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix} : a,b \in R \right\rbrace. \end{align} I guess that $I$ is either simple of semisimple as right $S$-module under the assumption that $R$ is semisimple.
How can I prove this claim?.
Thanks in advance.