= Localization away from one plus an ideal lies in the Jacobson radical
{title2=$(1+I)^{-1}I\subseteq J((1+I)^{-1}R)$}
For a <commutative ring> $R$ and <ideal> $I$, the set $S=1+I$ is a <multiplicative subset>. For $i\in I$ and $s,t\in S$, the numerator $st-ri$ of $1-(r/t)(i/s)$ again lies in $S$. Thus this difference is a <unit> in the <localization of a ring>. The <unit criterion for the Jacobson radical> proves $S^{-1}I\subseteq J(S^{-1}R)$. This requires no <Noetherian> hypothesis.
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