Ring with 0 only nilpotent element

• Oct 25th 2008, 04:59 AM
petter
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 .
• Oct 25th 2008, 09:39 AM
Opalg
Quote:

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).
• Oct 27th 2008, 10:19 AM
petter
Quote:

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 ?
• Oct 27th 2008, 10:43 AM
Opalg
Quote:

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

Multiply on the left by $a$.
• Oct 27th 2008, 01:12 PM
petter
Quote:

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

I'm stupid .
Thx !