Localization away from one plus an ideal lies in the Jacobson radical ($(1+I{)}^{-1}I\subseteq J((1+I{)}^{-1}R)$)

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.