I'm working through a proof of this statement which goes roughly as follows:
"The simple submodules of $R$ (as a module over $R$) are exactly the minimal left ideals of $R$. So (from earlier theorem) $R = \bigoplus_{i \in I} S_i$ where each $S_i$ is a minimal left ideal...''
Then comes the part that I don't understand:
"...In particular the element $1 \in R$ can be written as a finite sum, \begin{eqnarray*} 1 = x_{i_1} + \dots + x_{i_n} \end{eqnarray*} where $x_{i_j} \in S_{i_j}$. It then follows by multiplication of $r = r \cdot 1$...''
I don't see why $1$ can be written as this finite sum?