Problem
Characterize the semisimple rings $R$ that contain a unique maximal ideal.
I am not so sure what to do here. I know that a ring $R$ is semisimple if and only if all $R$-modules are semisimple. And I also know that if $I$ is a maximal ideal of $R$, then $R/I$ is isomorphic to $M$, where $M$ is a simple $R$-module. I don't know if this last information is of any help to solve the problem, any suggestions would be appreciated.
By the Artin-Wedderburn theorem, a semisimple ring is a direct product of simple rings, i.e. $R=\prod_{i=1}^n R_i$.
The maximal ideals are of the form $M_j=\prod_{i=1}^n I_i$ where $I_i=R_i$ for $i\neq j$ and $I_j=\{0\}$ for a particular $j$, $1\leq j\leq n$.
Thus the only semisimple rings with a unique maximal ideal are the simple ones.