Let $\mathscr{I}_\sigma$ be the Gabriel filter of ideals for a hereditary torsion theory $\sigma$ over a commutative ring $R$. I am looking for equivalent conditions on either $\sigma$ or $R$ under which for each idempotent element $e$ of $R$ either $\langle e\rangle\in \mathscr{I}_\sigma$ or $\langle 1-e\rangle\in \mathscr{I}_\sigma$.

Let $\mathscr{I}_\sigma$ be the Gabriel filter of ideals for a hereditary torsion theory $\sigma$ over a commutative ring $R$. I am looking for equivalent conditions on either $\sigma$ or $R$ under which for each idempotent element $e$ of $R$ either $\langle e\rangle\in \mathscr{I}_\sigma$ or $\langle 1-e\rangle\in \mathscr{I}_\sigma$.

Note that a Gabriel filter of ideals $\mathscr{I}_\sigma$ , a non–empty filter of ideals satisfying that every ideal $I$ of $R$, for which there exists an ideal $J\in\mathscr{I}_\sigma$ such that $(I : j):=\{r\in R\mid rj\in I\}\in\mathscr{I}_\sigma$ , for every $j\in J$, belongs to $\mathscr{I}_\sigma$ .