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 $\langle 1-e\rangle\in \mathscr{I}_\sigma$.