Show that R is a skew field.

Question

Let $R$ be a ring with unity such that the only left ideals of $R$ are $\{0\}$ and $R$. Show that $R$ is a skew field.

Please help me to solve this problem.

Answer 1

Let $a\neq 0$ .

Consider $Ra$ .(Show that it is a left ideal of $R$).

$a=1.a\in Ra\implies Ra\neq \{0\}$.

Conclusion $Ra=R\implies \exists b\in R$ such that $ba=1$.

To conclude the proof show that $ab=ba=1$(How?)

Also similarly $Rb=R\implies cb=1 $ for some $c\in R$.Now $a=1.a=(cb).a=c.(ba)=c.1=c$