# Ring with 0 only nilpotent element

• Oct 25th 2008, 03:59 AM
petter
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 .
• Oct 25th 2008, 08:39 AM
Opalg
Quote:

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).
• Oct 27th 2008, 09:19 AM
petter
Quote:

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 ?
• Oct 27th 2008, 09:43 AM
Opalg
Quote:

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

Multiply on the left by $\displaystyle a$.
• Oct 27th 2008, 12:12 PM
petter
Quote:

Originally Posted by Opalg
Multiply on the left by $\displaystyle a$.

I'm stupid .
Thx !