Let $\displaystyle A$ be a ring in wich $\displaystyle 0$ is the only nilpotent element ( $\displaystyle x^n=0 \rightarrow x=0 $ ) . Prove that $\displaystyle x$ is invertible if and only if x is left invertible .
Let $\displaystyle A$ be a ring in which $\displaystyle 0$ is the only nilpotent element ( $\displaystyle x^n=0 \rightarrow x=0 $ ) . Prove that $\displaystyle x$ is invertible if and only if x is left invertible .
Hint: Let $\displaystyle a$ be a left inverse for $\displaystyle x$. Then $\displaystyle x(1-xa)$ is nilpotent (just check that its square is 0).