Show that a division ring is simple.
With a division ring I mean a ring such that all onzero elements are invertible. And with a simple ring I mean a ring which has exactly two two-sided ideals.
I'm doing some exercises to prepare for a new course that start next week. I'm trying to get my ring theory neurons back in the shape they used to be 6 months ago
From here: If not, then any two sided ideal of a division ring $R$ has a nonzero element $r$. However, $Ra$ would then contain $1$. This is a contradiction.