 Non-Noetherian failure of the intersection formula for localization

Letbe the semigroup algebra inside $k[t^{\mathbb{Q}}]$, and let $I$ be generated by the $t^q$ with $q>0$. This integral domain is not Noetherian, sinceMoreover $I^2=I$, because $t^q=(t^{q/2}{)}^2$. Thus ${\bigcap }_{j\ge 1}{I}^j=I\ne 0$, whereas the map $R\to (1+I{)}^{-1}R$ is injective because $R$ is a domain.