# Every ring with $1$ and with no zero divisors and no non-trivial ideals is a division ring

It is well known that every commutative ring with unity $R$ that contains no non-trivial ideal is a field, since given $a \neq 0$, $(a)=R$, therefore there exists $x \in R$ with $ax=xa=1$. What happens if we remove the word commutative? Will it become a division ring?

The answer is false: If $K$ is a field, $M_n(K)$ $(n>1)$ has contains no non-trivial ideal but it's not a division ring.

But what happens if we also suppose the ring $R$ has no zero divisors?

• The correct requirement is that it has no nontrivial left (or right) ideals. Two-sided ideals are hard to come by in any case. – Matt Samuel Aug 27 '15 at 3:56