Ring with 0 only nilpotent element

**petter** · October 25th 2008, 03:59 AM

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 .

**Opalg** · October 25th 2008, 08:39 AM

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).

**petter** · October 27th 2008, 09:19 AM

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 ?

**Opalg** · October 27th 2008, 09:43 AM

Originally Posted by petter

So $x(1-xa)=0$ . How to continue ?

Multiply on the left by $a$ .

**petter** · October 27th 2008, 12:12 PM

Originally Posted by Opalg

Multiply on the left by $a$ .

I'm stupid .
Thx !

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