# Math Help - Ring with 0 only nilpotent element

1. ## Ring with 0 only nilpotent element

Let $A$ be a ring in wich $0$ is the only nilpotent element ( $x^n=0 \rightarrow x=0$ ) . Prove that $x$ is invertible if and only if x is left invertible .

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

3. Originally Posted by Opalg
Hint: Let $a$ be a left inverse for $x$. Then $x(1-xa)$ is nilpotent (just check that its square is 0).
So $x(1-xa)=0$ . How to continue ?

4. Originally Posted by petter
So $x(1-xa)=0$ . How to continue ?
Multiply on the left by $a$.

5. Originally Posted by Opalg
Multiply on the left by $a$.
I'm stupid .
Thx !