From Louis H. Rowen book Ring Theory, We have this proposition:
My first question:
In the proof, he just showed that every nonzero element has a left inverse but to prove the ring is a division ring we need also to show that every nonzero element has a right inverse. So, why does every nonzero element has a right inverse here?
My second question: in the same book, we have this proposition
but I am confused here because the first proposition above stated that it is possible that there is a division ring with a proper nonzero left ideal but this is a contradiction with proposition 2.0.16 above?
Suppose that $x,y,z \in D$, where $D$ is a ring and $zy=yx=1$ (so that $z$ is a left inverse of $y$, which is in turn a left inverse of $x$). Then, $x=1x=(zy)x=z(yx)=z1=z$, so in fact $xy=yx=1$ and $y$ is then a two-sided inverse of $x$.
In fact, the above argument works in any monoid.