# Show that a division ring is simple.

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

• And where are you stuck at? Hint: a division ring $D$ has no left ideal different from $\{0\}$ and $D$. – egreg Aug 31 '14 at 16:32
• Hint - choose an ideal. If it is not the zero ideal it has a non-zero element. Use what you know together with the property of being an ideal to reduce the possibilities. – Mark Bennet Aug 31 '14 at 16:32
• @MarkBennet oh wait.... like if $x$ in the ideal, then $x^{-1}\cdot x$ in the ideal, so everything in the ideal – 90intuition Aug 31 '14 at 16:36

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.