Proving 1+x is a Unit in a Ring for x^n=0

[ksk104]

Homework Statement

Let R be a ring and x in R such that x^n=0 for some n show that 1 + x is a unit.

I know then that x is a zero divisor and I need to find y such that y(1+x) = 1.
I can see in examples that this works and I can prove it for Z mod n. I can't figure out how to prove it for any ring. Please help
Thanks

 

[HallsofIvy]

If there exist n such that xⁿ, there exist a smallest such n. Assume, without loss of generality that n is the smallest number such that xⁿ= 0. If n= 1, then x= 0, x+1= 1 which is a unit. If n> 1, then x^n-1 is not 0. Let u= x^n-1. Then u(1+ x)= x^n-1+ xⁿ= x^n-1= u. Does that lead anywhere? In particular is u a unit?

 

[Dick]

You probably know that you can write 1/(1+x) formally as a power series, 1-x+x^2-x^3+... If x^n=0, that series terminates. Can you show that it's true that (1+x)*(1-x+x^2-...x^(n-1))=1?