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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?

rschwieb
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zacarias
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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. – Andrew Maurer Dec 20 '12 at 16:02
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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. – zacarias Dec 20 '12 at 16:15
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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... – tomasz Dec 20 '12 at 16:20
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    Note: The question is asking for a reference for the above fact, not a proof. We already have questions (here, here) regarding how to prove this. – Zev Chonoles Dec 20 '12 at 17:19

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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.

Ted
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