Thread: Ring with 0 only nilpotent element

1. Ring with 0 only nilpotent element

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 .

2. Originally Posted by petter
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).

3. Originally Posted by Opalg
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).
So $\displaystyle x(1-xa)=0$ . How to continue ?

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

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