I was doing some reading and I came across the following statement:
"Let $R$ be a ring and let $I$ be a two-sided ideal of $R$ such that $R/I$ is semisimple. Then $I$ is a finite intersection of maximal ideals $M \subset R$ such that $R/M$ is simple Artinian. Therefore, we may as well assume that $R/I$ is simple Artinian."
I have difficulties understanding this statement. Does anyone care to explain?
A semisimple ring $\prod_{i=1}^n R_i$ (each $R_i$ simple Artinian) has exactly $n$ maximal ideals, one for each coordinate. They are all of the form $\prod_{i=1}^n I_i$ where $I_j=\{0\}$ for one $j$ and $I_i=R_i$ for $i\neq j$.
Now, the maximal ideals of $R/I$ correspond exactly to the maximal ideals of $R$ containing $I$. Therefore there are $n$ maximal ideals of $R$ containing $I$, and their intersection is $I$.
I do not know precisely what the context is, so I cannot comment on why it is acceptable to handle the case where $R/I$ is simple Artinian. You would have to provide more background information.