Hi

Let $\displaystyle A$ be a ring, $\displaystyle a,b \in A$ such that $\displaystyle ab=1$. Suppose that $\displaystyle X=\{x \in A;\ ax=0\}$ is finite.

Prove that $\displaystyle ba=1$

We can see that $\displaystyle 1-ba \in X$, so if $\displaystyle X=\{0\}$, it's over.

Any idea?