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# Abstract algebra question watch

1. Can some one tell me if it is true that: in a commutative ring A, if we have a an nilpotent element and x a reversible element is it true that ax^(-1)+1 is reversible ???
2. (Original post by mariucell)
Can some one tell me if it is true that: in a commutative ring A, if we have a an nilpotent element and x a reversible element is it true that ax^(-1)+1 is reversible ???
If by reversible you mean invertible, then yes this is true.
3. (Original post by Mark13)
If by reversible you mean invertible, then yes this is true.
Yesss, that I mean. Thanks for the answer, do you know any proof of this. I am a student in the final year of high school from Romania and I don't know if it known by many. Saw it on a paper quoted without demonstration and I only see some particular cases, not the general property . Does it involve binomial theorem ?
4. (Original post by mariucell)
Yesss, that I mean. Thanks for the answer, do you know any proof of this. I am a student in the final year of high school from Romania and I don't know if it known by many. Saw it on a paper quoted without demonstration and I only see some particular cases, not the general property . Does it involve binomial theorem ?
If a is nilpotent, say a^n=1, then the inverse of 1+a is 1-a+a^2-a^3+...+(-a)^(n-1). You can check this just by multiplying 1+a by this element.

If a is nilpotent and x is invertible, then (in a commutative ring) ax^(-1) is nilpotent, so you can use the same idea.
5. Thanks a lot, a bit disappointed for not seeing this although worked something close to this . Thanks again

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