Let $R$ be a commutative ring. Prove that the element $x\in R$ is a unit iff $(x+RadR)$ is a unit in $R/RadR$. ($RadR$ is Jacobson radical of $R$)
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$\hat x$ is invertible in $R/J(R)$ implies $\hat x\hat y=\hat 1$ for some $y\in R$, that is, $1-xy\in J(R)$, so $1-(1-xy)=xy$ is invertible (in $R$) and therefore $x$ is invertible.
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Can you explain why $xy$ is invertible in $R$ implies $x$ is invertible? – Truong Nov 23 '13 at 23:41
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Okay. Thank you very much. – Truong Nov 23 '13 at 23:43
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This post makes your hint more precise http://math.stackexchange.com/questions/99949/if-xy-is-a-unit-are-x-and-y-units?rq=1 – Truong Nov 23 '13 at 23:49