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Let $R$ be a commutative ring with identity and denote by $\mathcal N(R)$ its nilradical. It is known that an element $u\in R$ is a unit if and only if $u+x$ is a unit for all $x\in\mathcal N(R)$. In which book I can find a proof of this fact?

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    Why don't you try to work this out yourself? It's not terribly difficult and should be a good exercise.2012-12-20
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    I need to mention this in a paper, but I have not space in the paper to present a proof. So, I need to make reference in the paper.2012-12-20
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    @zacarias: I'd think that in a scientific paper you can cite facts as basic as this without any proof or reference...2012-12-20
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    **Note:** *The question is asking for a reference for the above fact, not a proof*. We already have questions ([here](http://math.stackexchange.com/q/119904/264), [here](http://math.stackexchange.com/q/151856/264)) regarding how to prove this.2012-12-20

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Expand $\frac{1}{u+x}$ as a geometric series (equivalently, Taylor series) in $x$. Since $x$ is nilpotent, the series terminates and there are no convergence issues.