1
$\begingroup$

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 $\mathscr{I}_\sigma$ .

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ I suspect that the notation and terminology you're using might not be defined in exactly the same way by all authors. For the benefit of those reading (and potentially answering) the question, it might be worthwhile to remind us of some of the definitions (such as the notion of a Gabriel filter, perhaps even the notion of a hereditary torsion theory) and notation (such as the angle brackets), or else cite / link to places where we can brush up. It might also not hurt to provide a bit of context or an example of an $R$ you care about, especially since the question is a bit open-ended. $\endgroup$
    – Tim Campion
    Commented Jan 10, 2022 at 20:17

0

Reset to default

You must log in to answer this question.